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In this paper, we develop a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for simulating the multicomponent, chemically reacting, compressible Euler equations with complex thermodynamics. The proposed formulation is an extension of the fully conservative, high-order numerical method previously developed by Johnson and Kercher (2020) [14] that maintains pressure equilibrium between adjacent elements. In this first part of our two-part paper, we focus on the one-dimensional case. Our methodology is rooted in the minimum entropy principle satisfied by entropy solutions to the multicomponent, compressible Euler equations, which was proved by Gouasmi et al. (2020) [16] for nonreacting flows. We first show that the minimum entropy principle holds in the reacting case as well. Next, we introduce the ingredients, including a simple linear-scaling limiter, required for the discrete solution to have nonnegative species concentrations, positive density, positive pressure, and bounded entropy. We also discuss how to retain the aforementioned ability to preserve pressure equilibrium between elements. Operator splitting is employed to handle stiff chemical reactions. To guarantee discrete satisfaction of the minimum entropy principle in the reaction step, we develop an entropy-stable discontinuous Galerkin method based on diagonal-norm summation-by-parts operators for solving ordinary differential equations. The developed formulation is used to compute canonical one-dimensional test cases, namely thermal-bubble advection, advection of a low-density Gaussian wave, multicomponent shock-tube flow, and a moving hydrogen-oxygen detonation wave with detailed chemistry. We demonstrate that the formulation can achieve optimal high-order convergence in smooth flows. Furthermore, we find that the enforcement of an entropy bound can considerably reduce the large-scale nonlinear instabilities that emerge when only the positivity property is enforced, to an even greater extent than in the monocomponent, calorically perfect case. Finally, mass, total energy, and atomic elements are shown to be discretely conserved. [ABSTRACT FROM AUTHOR]

In this work, a bound- and positivity-preserving quasi-conservative discontinuous Galerkin (DG) method is proposed for the γ -based model of compressible two-medium flows. The contribution of this paper mainly includes three parts. On one hand, the DG method with the extended Harten-Lax-van Leer contact flux is proposed to solve the γ -based model, and satisfies the equilibrium-preserving property which preserves uniform velocity and pressure fields at an isolated material interface. On the other hand, an affine-invariant weighted essentially non-oscillatory (Ai-WENO) limiter is adopted to suppress oscillations near the discontinuities. The limiter with the Ai-WENO reconstruction method to the conservative variables not only is able to maintain the equilibrium property, but also generates sharper results around the locations of shock waves in contrast to that applying to the primitive variables. Last but not least, a flux-based bound- and positivity-preserving limiting strategy is introduced and analyzed, which preserves the physical bounds for auxiliary variables in the non-conservative governing equations, and the positivity for density and internal energy. Extensive numerical experiments in both one and two space dimensions show that the proposed method performs well in simulating compressible two-medium flows with high-order accuracy, equilibrium-preserving and bound-preserving properties. [ABSTRACT FROM AUTHOR]

In this paper, we develop high order discontinuous Galerkin (DG) methods with Lax-Friedrich fluxes for Euler equations under gravitational fields. Such problems may yield steady-state solutions and the density and pressure are positive. There were plenty of previous works developing either well-balanced (WB) schemes to preserve the steady states or positivity-preserving (PP) schemes to get positive density and pressure. However, it is rather difficult to construct WB PP schemes with Lax-Friedrich fluxes, due to the penalty term in the flux. In fact, for general PP DG methods, the penalty coefficient must be sufficiently large, while the WB scheme requires that to be zero. This contradiction can hardly be fixed following the original design of the PP technique, where the numerical fluxes in the DG scheme are treated separately. However, if the numerical approximations are close to the steady state, the numerical fluxes are not independent, and it is possible to use the relationship to obtain a new penalty parameter which is zero at the steady state and the full scheme is PP. To be more precise, we first reformulate the source term such that it balances with the flux term when the steady state has reached. To obtain positive numerical density and pressure, we choose a special penalty coefficient in the Lax-Friedrich flux, which is zero at the steady state. The technique works for general steady-state solutions with zero velocities. Numerical experiments are given to show the performance of the proposed methods. • Construct positivity-preserving and well-balanced schemes for Euler equations with gravitation. • The penalty is chosen to be zero in the Lax-Friedrichs fluxes at the steady-states. • A novel positivity-preserving technique is developed. • The scheme is straightforward to implement. [ABSTRACT FROM AUTHOR]

In this second part of our two-part paper, we extend to multiple spatial dimensions the one-dimensional, fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme developed in the first part for the chemically reacting Euler equations. Our primary objective is to enable robust and accurate solutions to complex reacting-flow problems using the high-order discontinuous Galerkin method without requiring extremely high resolution. Variable thermodynamics and detailed chemistry are considered. Our multidimensional framework can be regarded as a further generalization of similar positivity-preserving and/or entropy-bounded discontinuous Galerkin schemes in the literature. In particular, the proposed formulation is compatible with curved elements of arbitrary shape, a variety of numerical flux functions, general quadrature rules with positive weights, and mixtures of thermally perfect gases. Preservation of pressure equilibrium between adjacent elements, especially crucial in simulations of multicomponent flows, is discussed. Complex detonation waves in two and three dimensions are accurately computed using high-order polynomials. Enforcement of an entropy bound, as opposed to solely the positivity property, is found to significantly improve stability. Mass, total energy, and atomic elements are shown to be discretely conserved. [ABSTRACT FROM AUTHOR]

In this paper we develop a new well-balanced discontinuous Galerkin (DG) finite element scheme with subcell finite volume (FV) limiter for the numerical solution of the Einstein–Euler equations of general relativity based on a first order hyperbolic reformulation of the Z4 formalism. The first order Z4 system, which is composed of 59 equations, is analyzed and proven to be strongly hyperbolic for a general metric. The well-balancing is achieved for arbitrary but a priori known equilibria by subtracting a discrete version of the equilibrium solution from the discretized time-dependent PDE system. Special care has also been taken in the design of the numerical viscosity so that the well-balancing property is achieved. As for the treatment of low density matter, e.g. when simulating massive compact objects like neutron stars surrounded by vacuum, we have introduced a new filter in the conversion from the conserved to the primitive variables, preventing superluminal velocities when the density drops below a certain threshold, and being potentially also very useful for the numerical investigation of highly rarefied relativistic astrophysical flows. Thanks to these improvements, all standard tests of numerical relativity are successfully reproduced, reaching three achievements: (i) we are able to obtain stable long term simulations of stationary black holes, including Kerr black holes with extreme spin, which after an initial perturbation return perfectly back to the equilibrium solution up to machine precision; (ii) a (standard) TOV star under perturbation is evolved in pure vacuum (ρ = p = 0) up to t = 1000 with no need to introduce any artificial atmosphere around the star; and, (iii) we solve the head on collision of two punctures black holes, that was previously considered un–tractable within the Z4 formalism. Due to the above features, we consider that our new algorithm can be particularly beneficial for the numerical study of quasi normal modes of oscillations, both of black holes and of neutron stars. • Strongly hyperbolic first order Z4 formulation of the Einstein-Euler equations. • New and simple well-balanced discontinuous Galerkin schemes for numerical general relativity. • Robust conversion from conservative to primitive variables also in the presence of vacuum. • Stable long-time simulations of black holes and TOV stars in two and three space dimensions. • Head-on collision of two puncture black holes in three space dimensions using the Z4 formulation. [ABSTRACT FROM AUTHOR]

The numerical approximation of the shallow-water equations, which support geophysical wave propagation, namely the fast external Poincaré and Kelvin waves and the slow large-scale planetary Rossby waves, is a delicate and difficult problem. Indeed, the coupling between the momentum and the continuity equations may lead to the presence of erratic or spurious solutions for these waves, e.g. the spurious pressure and inertial modes. In addition, the presence of spurious branches and the emergence of spectral gaps in the dispersion relation at specific wavenumbers may lead to anomalous dissipation/dispersion in the representation of both fast and slow waves. The aim of the present study is to propose a class of possible discretization schemes, via the discontinuous Galerkin method, that is not affected by the above-mentioned problems. A Fourier/stability analysis of the 2D shallow-water model is performed by analysing the finite volume method and the linear discontinuous P 1 D G and non conforming P 1 N C Galerkin discretizations. These are stabilized by means of a family of numerical fluxes via the Polynomial Viscosity Matrix approach. A long time stability result is proven for all schemes and fluxes examined here. Further, a super-convergent result is demonstrated for the P 1 N C discrete frequencies compared to the P 1 D G ones, except for the slow mode in the Roe flux case. Indeed, we show that the Roe flux yields spurious frequencies and sub-optimal rates of convergence for the slow mode, for both the P 1 D G and P 1 N C methods. Finally, numerical solutions of linear and non-linear test problems confirm the theoretical results and reveal the computational cost efficiency of the P 1 N C approach. • A Fourier analysis of the 2D shallow-water model is performed for the linear DG and NC Galerkin discretizations. • A long time stability result is proven for all schemes and fluxes examined in this paper. • A super-convergent result is demonstrated for the NC frequencies compared to the DG ones. • We show that the Roe flux yields spurious frequencies in O(h) for the Rossby frequency for both the DG and NC methods. • The CFL number of the NC method is 5/3 as large as that of the DG method. [ABSTRACT FROM AUTHOR]

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov–Poisson–Lenard–Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0 x 3 v space homogeneous geometry and a 1 x 3 v slab geometry. We use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD. [ABSTRACT FROM AUTHOR]

In this paper, we propose and numerically solve a model considering confined flow in fractured porous media coupled with free flow in conduits. Such a situation arises, for example, for fluid flows in fractured vuggy carbonate reservoirs and caves/conduits, where a matrix and fractures are the main spaces for fluid storage and flow. The flow in caves and conduits is governed by the Stokes equation; the flow in fractured porous media, which consists of both a matrix and fractures, is described by a coupled Darcy–Darcy (bulk–fracture) model. Then, the two models are coupled through suitable (physically consistent) interface conditions on the interface between fractured porous media and caves/conduits, playing a key role in a physically faithful simulation that achieves high accuracy. The construction of all the interface conditions of the coupled model is based on the fundamental properties of the traditional bulk–fracture model and the well-known Stokes–Darcy model. The weak formulation is derived for the proposed model, and the well-posedness of the model is theoretically analyzed. We also propose a polyhedral discontinuous Galerkin finite element approximation for the Stokes flow in conduits and Darcy flow in the porous matrix, which is coupled with a conforming finite element scheme for the flow in the fracture. Optimal error estimates in a suitable energy norm are obtained. Two-dimensional numerical experiments are conducted to validate the proposed model and to demonstrate the features of both the model and the numerical method, such as the optimal convergence rate of the numerical solution; the detail flow characteristics around fractures, the matrix, and conduits; and the applicability to real-world problems. [ABSTRACT FROM AUTHOR]

In this paper, a selective discontinuous Galerkin (SDG) method and a polynomial preserving recovery (PPR) method are developed and integrated with the immersed-finite-element particle-in-cell (IFE-PIC) method, in order to carry out the plasma-material interaction simulation on adaptive Cartesian meshes (ACM). To significantly save the computational cost in practice, the PIC simulation often needs to use Cartesian meshes for the whole domain and the ACM for some focused regions in the plasma-material interaction problems. Therefore, we utilize the SDG method with immersed finite elements for the field solver of the IFE-PIC method, as the key technique to allow the local Cartesian mesh refinement which will generate hanging nodes. To minimize the degree of freedom of the SDG on the ACM and reduce the computational cost, a new selective technique of the discontinuous global basis functions is proposed for the SDG. Various types of nodes in the ACM are discussed for the implementation of this selective technique, including the hanging nodes. Meanwhile, the gathering and scattering steps of the PIC simulation also need to be appropriately performed on the ACM. The PPR method is a generic gradient recovery technique to be incorporated into the IFE-PIC method for more accurate force deposition on the ACM. In addition, two other algorithmic structures of the traditional IFE-PIC method have been modified to accommodate the ACM in the simulation, including a new mapping array structure for the particle positioning and a new charge deposition with some special particle positions in the ACM. As a result, the integrated immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method can efficiently perform the plasma-material interactions simulation on the Cartesian meshes with local mesh refinement independent of the interface. Several numerical experiments are performed to demonstrate the convergence, efficiency, and applicability of the proposed ISDG-PIC method. • Immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method. • Polynomial preserving recovery (PPR) method. • A new mapping array structure for the particle positioning. • A new charge deposition with some special particle positions in the adaptive Cartesian meshes. • A series of numerical examples for validation. [ABSTRACT FROM AUTHOR]

This paper proposes a hybrid limiter in discontinuous Galerkin (DG) methods for hyperbolic conservation laws. This hybrid limiter combines the high-order DG approximation with the low-order limited solution to achieve better resolution for multiscale structures. Meanwhile, the essentially non-oscillatory property can also be retained. The indicator, based on the jump information at the cell interface, provides the non-linear weight for the hybrid limiter, which also acts as a shock detection technique to identify troubled cells in the DG approximation. The presented limiter also maintains the conservation of mass cell by cell. In addition, the hybrid limiter maintains the local data structure and can, therefore, preserve the parallel efficiency of the DG method. The hybrid limiter is very simple in design and can be easily implemented with arbitrary high-order accuracy on both structured and unstructured meshes. Benchmark examples, mainly on Euler equations, are presented to demonstrate the performance of these DG methods with associated hybrid limiters. [ABSTRACT FROM AUTHOR]

Recently, there has been a series of works on the positivity-preserving discontinuous Galerkin methods for stationary hyperbolic equations, where the notion of mass conservation follows from a straightforward analogy of that of time-dependent problems, i.e. conserving the mass = preserving cell averages during limiting. Based on such a notion, the implementations and theoretical proofs of positivity-preserving limited methods for stationary equations are unnecessarily complicated and constrained. As will be shown in this paper, in some extreme cases, their convergence could even be problematic. In this work, we clarify a more appropriate definition of mass conservation for limiters applied to stationary hyperbolic equations and establish the genuinely conservative high-order positivity-preserving limited discontinuous Galerkin methods based on this definition. The new methods are able to preserve the positivity of solutions of scalar linear equations and scalar nonlinear equations with invariant wind direction, with much simpler implementations and easier proofs for accuracy and the Lax-Wendroff theorem, compared with the existing methods. Two types of positivity-preserving limiters preserving the local mass of stationary equations are developed to accommodate for the new definition of conservation and their accuracy are investigated. We would like to emphasize that a major advantage of the original DG scheme presented in [24] is a sweeping procedure, which allows for the computation of conservative steady-state solutions explicitly, cell by cell, without iterations, even for nonlinear equations as long as the wind direction is fixed. The main contribution of this paper is to introduce a limiting procedure to enforce positivity without changing the conservative property of this original DG scheme. The good performance of the algorithms for stationary hyperbolic equations and their applications in time-dependent problems are demonstrated by ample numerical tests. • A new definition of local conservation is given for stationary hyperbolic systems. • This allows the design of positivity-preserving discontinuous Galerkin (DG) schemes in more general cases than before. • Such high order positivity-preserving DG schemes are more general than before. [ABSTRACT FROM AUTHOR]

• Theoretical comparison between hypoelastic and hyperelastic description of the dynamics of solids. • Wilkins model for hypoelasticity and Godunov-Peshkov-Romenski model for hyperelasticity. • Numerical comparison of both model using FV and DG schemes on structured and unstructured meshes. The aim of this paper is to compare a hyperelastic with a hypoelastic model describing the Eulerian dynamics of solids in the context of non-linear elastoplastic deformations. Specifically, we consider the well-known hypoelastic Wilkins model, which is compared against a hyperelastic model based on the work of Godunov and Romenski. First, we discuss some general conceptual differences of the two approaches. Second, a detailed study of both models is proposed, where differences are made evident at the aid of deriving a hypoelastic-type model corresponding to the hyperelastic model and a particular equation of state used in this paper. Third, using the same high order ADER Finite Volume and Discontinuous Galerkin methods on fixed and moving unstructured meshes for both models, a wide range of numerical benchmark test problems has been solved. The numerical solutions obtained for the two different models are directly compared with each other. For small elastic deformations, the two models produce very similar solutions that are close to each other. However, if large elastic or elastoplastic deformations occur, the solutions present larger differences. [ABSTRACT FROM AUTHOR]

Highlights • Construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. • Treat the time derivative of the pressure as a source of the concentration equation. • Apply the algorithm on unstructured meshes. • In the flux limiter, use the second-order flux as the lower-order one. • Use L2-projection of the porosity and construct special limiters that suitable for multi-component fluid mixtures. Abstract In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the j th component, c j , should be between 0 and 1. There are three main difficulties. Firstly, c j does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in Zhang and Shu (2010) [44] cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all c j ′ s and enforce ∑ j c j = 1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure d p / d t as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to c j. To construct the BP technique, we will not approximate c j directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique. [ABSTRACT FROM AUTHOR]

Abstract This paper develops a discontinuous Galerkin method to compute a few smallest elasticity transmission eigenvalues, which are of practical importance in inverse elastic scattering theory. For high-order problems, comparing with classical conforming finite element methods, discontinuous Galerkin methods use simpler basis functions which make the numerical implementation much easier. In this paper, we propose an interior penalty discontinuous Galerkin method using C 0 Lagrange elements (C 0 IP) for the elastic transmission eigenvalue problem and prove the optimal convergence. Numerical examples are presented to validate its effectiveness. Both real and complex eigenvalues can be obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel interior penalty discontinuous Galerkin projection method for the incompressible magneto-hydrodynamic equations. The scheme is employed by an implicit-explicit treatment of the nonlinear coupling terms and a second-order rotational pressure-correction scheme for dealing with the Navier-Stokes equations. One noteworthy aspect of this scheme is the introduction of an additional stabilization term to Maxwell's equations, which allows for the explicit treatment of the coupled nonlinear terms and the decoupling of computations for the magnetic and velocity fields, ultimately leading to the achievement of desired linearity, full decoupling, second-order accuracy in time, and unconditional energy stability. The obtained scheme is easy to implement because one only needs to solve a few decoupled linear equations at each time step. We rigorously prove the unique solvability and unconditional energy stability of the developed scheme and present a series of numerical examples to demonstrate the accuracy, stability, and efficiency of the proposed scheme. • A novel interior penalty discontinuous Galerkin projection scheme for the MHD system is developed. • The scheme is linear, fully-decoupled, second-order in time and unconditional energy stable. • The well-posedness and energy stability of the scheme are rigorously proved. • Extensive numerical experiments are provided to show the efficiency and good performance of the method. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in constructing a scheme solving compressible Navier–Stokes equations, with desired properties including high order spatial accuracy, conservation, and positivity-preserving of density and internal energy under a standard hyperbolic type CFL constraint on the time step size, e.g., Δ t = O (Δ x). Strang splitting is used to approximate convection and diffusion operators separately. For the convection part, i.e., the compressible Euler equation, the high order accurate postivity-preserving Runge–Kutta discontinuous Galerkin method can be used. For the diffusion part, the equation of internal energy instead of the total energy is considered, and a first order semi-implicit time discretization is used for the ease of achieving positivity. A suitable interior penalty discontinuous Galerkin method for the stress tensor can ensure the conservation of momentum and total energy for any high order polynomial basis. In particular, positivity can be proven with Δ t = O (Δ x) if the Laplacian operator of internal energy is approximated by the Q k spectral element method with k = 1 , 2 , 3. So the full scheme with Q k (k = 1 , 2 , 3) basis is conservative and positivity-preserving with Δ t = O (Δ x) , which is robust for demanding problems such as solutions with low density and low pressure induced by high-speed shock diffraction. Even though the full scheme is only first order accurate in time, numerical tests indicate that higher order polynomial basis produces much better numerical solutions, e.g., better resolution for capturing the roll-ups during shock reflection. • The hyperbolic subproblem is solved by a positivity-preserving RKDG method. • The parabolic subproblem is solved by classical schemes to achieve positivity. • A proper choice of the IPDG method achieves total conservation. • Numerically a spatially higher order scheme produces better solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier–Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order rotational incremental pressure correction scheme. The major focus of the paper is threefold: i) We propose a modified upwind scheme based on the Vijayasundaram numerical flux that has favourable properties in the context of DG. ii) We present a novel postprocessing technique in the Helmholtz projection step based on H ( div ) reconstruction of the pressure correction that is computed locally, is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly. As a consequence it also provides local mass conservation of the projected velocity. iii) Numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two-dimensional problems with known solution as well as large-scale three-dimensional problems. In particular we address second-order convergence in time of the splitting scheme as well as its long-time stability. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to constraining the eigenvalue range of symmetric tensors in numerical advection schemes based on the flux-corrected transport (FCT) algorithm and a continuous finite element discretization. In the context of element-based FEM-FCT schemes for scalar conservation laws, the numerical solution is evolved using local extremum diminishing (LED) antidiffusive corrections of a low order approximation which is assumed to satisfy the relevant inequality constraints. The application of a limiter to antidiffusive element contributions guarantees that the corrected solution remains bounded by the local maxima and minima of the low order predictor. The FCT algorithm to be presented in this paper guarantees the LED property for the maximal and minimal eigenvalues of the transported tensor at the low order evolution step. At the antidiffusive correction step, this property is preserved by limiting the antidiffusive element contributions to all components of the tensor in a synchronized manner. The definition of the element-based correction factors for FCT is based on perturbation bounds for auxiliary tensors which are constrained to be positive semidefinite to enforce the generalized LED condition. The derivation of sharp bounds involves calculating the roots of polynomials of degree up to 3. As inexpensive and numerically stable alternatives, limiting techniques based on appropriate estimates are considered. The ability of the new limiters to enforce local bounds for the eigenvalue range is confirmed by numerical results for 2D advection problems. [ABSTRACT FROM AUTHOR]

This paper presents a Discontinuous Galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro-thermal problems. In this paper we discuss the fundamental equations for the transport of electricity and heat, in terms of macroscopic variables such as temperature and electric potential. A fully coupled nonlinear weak formulation for electro-thermal problems is developed based on continuum mechanics equations expressed in terms of energetically conjugated pair of fluxes and fields gradients. The weak form can thus be formulated as a Discontinuous Galerkin method. The existence and uniqueness of the weak form solution are proved. The numerical properties of the nonlinear elliptic problems i.e., consistency and stability, are demonstrated under specific conditions, i.e. use of high enough stabilization parameter and at least quadratic polynomial approximations. Moreover the prior error estimates in the H 1 -norm and in the L 2 -norm are shown to be optimal in the mesh size with the polynomial approximation degree. [ABSTRACT FROM AUTHOR]

In this work we propose a variational approach with cell-to-point Galerkin projections for studying two-phase interface advection problems dominated by surface tension. A Volume Of Fluid (VOF) algorithm is used for tracking and locating the evolution of the two-phase interface on a Cartesian grid and a finite element numerical scheme for solving the velocity-pressure state. The velocity field that drives the evolution of this interface is computed from the weak form of the Navier-Stokes equation where the surface tension force is represented in variational form by the continuous surface force (CSF) and continuous surface stress (CSS) methods. Standard numerical approaches solve the strong form of the Navier-Stokes equations and define the CSS term by taking the divergence of the surface tension tensor. This computation of the divergence term results in a singular force which is difficult to compute when the grid is refined since the tensor is computed in a discontinuous cell-by-cell way. In this work we use the variational formulation of the Navier-Stokes equation and avoid differentiation. The tensor, which is a function of the unit normal, is evaluated over regular Sobolev spaces by using a cell-to-point Galerkin projection. This allows a regular piece-wise continuous representation of the surface tensor and the unit normal based on the VOF reconstruction. In standard approaches the CSF surface force is computed by using the curvature, which is the divergence of the unit normal. In this paper we recover the curvature with point-wise Galerkin projection avoiding direct differentiation. Tests on convergence for two and three-dimension in the static and dynamical cases are reported to show the correct representation in the desired spaces. This method is also natural for coupling non uniform grid computation of the fluid with Cartesian grid of the VOF algorithm. • VOF simulation with variational formulation and Galerkin interpolation projection. • Surface tension term computed via CSS method with no direct differentiation. • Surface tension term computed via CSF method by using an extended curvature. • Tests for two and three-dimensional geometries in the static and dynamical cases. [ABSTRACT FROM AUTHOR]

• High order energy conserving DG methods which converge in optimal order are designed for linear, symmetric hyperbolic systems. • These DG methods are compared with upwind DG methods and energy conserving DG methods with central fluxes to demonstrate their superior performance especially for long time simulation. • Arbitrary high-order Lax-Wendroff type, energy-conserving time stepping methods are developed. We propose energy-conserving discontinuous Galerkin (DG) methods for symmetric linear hyperbolic systems on general unstructured meshes. Optimal a priori error estimates of order k + 1 are obtained for the semi-discrete scheme in one dimension, and in multi-dimensions on Cartesian meshes when tensor-product polynomials of degree k are used. A high-order energy-conserving Lax-Wendroff time discretization is also presented. Extensive numerical results in one dimension, and two dimensions on both rectangular and triangular meshes are presented to support the theoretical findings and to assess the new methods. One particular method (with the doubling of unknowns) is found to be optimally convergent on triangular meshes for all the examples considered in this paper. The method is also compared with the classical (dissipative) upwinding DG method and (conservative) DG method with a central flux. It is numerically observed for the new method to have a superior performance for long time simulations. [ABSTRACT FROM AUTHOR]

In this paper, we develop the fully discrete local discontinuous Galerkin (LDG) methods coupled with the implicit-explicit (IMEX) multistep time marching for solving the nonlinear Cahn-Hilliard equation. To be more specific, we rewrite the Cahn-Hilliard equation in a novel form by adding and subtracting a "linear" term. Then we discretize the spatial derivatives with the LDG methods and the temporal derivative with the IMEX multistep method. Finally, a series of numerical experiments are given to verify the accuracy, efficiency and validness of proposed method. [ABSTRACT FROM AUTHOR]

Coherent synchrotron radiation (CSR) remains a computationally difficult aspect in the numerical modeling of modern electron accelerators. In this paper, we present the theory and numerical implementation of a high-order discontinuous Galerkin method for computing CSR wake fields in a bunch compressor while including geometrical effects. We incorporate this method in our code: CSRDG (Coherent Synchtrotron Radiation with Discontinuous Galerkin). Using this code, wake potentials and associated quantities are calculated for the simple case of an electron bunch propagating along a single bend structure as well as for a complete bunch compressor chamber of the European XFEL. • Coherent synchrotron radiation is modeled with Maxwell's equations in time-domain. • Curvilinear coordinate and Fourier series transformations reduce problem complexity. • A Discontinuous Galerkin method used in solving the transformed Maxwell's equations. • Wake fields are computed for a European XFEL bunch compressor model. [ABSTRACT FROM AUTHOR]

An accurate modeling of the optical interactions in metallic nanostructures with subnanometer features requires an accurate description of quantum effects at the scale of billions of atoms. At such scale, first-principle methods are not computationally viable. Quantum hydrodynamic theory (QHT) has emerged as a powerful method that includes nonlocal contributions of the kinetic energy and the spatial dependence of the electron density, and it can predict both plasmon energy and spill-out effects in large metal nanoparticles. In this paper, we introduce a hybridizable discontinuous Galerkin method for solving Maxwell's equations coupled with a QHT model in order to account for quantum effects in three-dimensional metallic nanostructures. The coupled system of Maxwell's equations and QHT model is not only nonlinear but also multi-scale due to the interaction between the micrometer electromagnetic waves and the nanometer cavities of metallic nanostructures. We present extensive numerical experiments to validate the QHT model and demonstrate the capability of the HDG method to provide accurate solutions in the presence of strong nonlinearities and multiple length scales. These results offer a possibility to enhance nonlinear optical effects or to harness quantum mechanical electron tunneling by engineering metallic nanostructures at the quantum level. • An HDG method to simulate the equilibrium equation of quantum hydrodynamic theory. • An HDG method to simulate the first-order linear system of quantum hydrodynamic theory coupled with Maxwell's equations. • 2D simulation of metal nanowire and full 3D simulation of a periodic nanocoax array accounting for quantum effects. • Comparison of the optical response using quantum effects to that of local and nonlocal electron models for both applications. [ABSTRACT FROM AUTHOR]

Liouville's equation describes light propagation through an optical system. It governs the evolution of an energy distribution on phase space. This energy distribution is discontinuous across optical interfaces. Curved optical interfaces manifest themselves as moving boundaries on phase space. In this paper, an ADER discontinuous Galerkin (DG) method on a moving mesh is applied to solve Liouville's equation. In the ADER approach a temporal Taylor series is computed by replacing temporal derivatives with spatial derivatives using the Cauchy-Kovalewski procedure. The result is a fully discrete explicit scheme of arbitrary high order of accuracy. A moving mesh is not sufficient to be able to solve Liouville's equation numerically for the optical systems considered in this article. To that end, we combine the scheme with a new method we refer to as sub-cell interface method. When dealing with optical interfaces in phase space, non-local boundary conditions arise. These are incorporated in the DG method in an energy-preserving manner. Numerical experiments validate energy-preservation up to machine precision and show the high order of accuracy. Furthermore, the DG method is compared to quasi-Monte Carlo ray tracing for two examples showing that the DG method yields better accuracy in the same amount of computational time. • Explicit Taylor expansion on moving mesh for discontinuous Galerkin. • Non-local boundary conditions on phase space. • High order energy-preserving method for geometrical optics. • Outperforms quasi-Monte Carlo ray tracing in a few examples. [ABSTRACT FROM AUTHOR]

The goal of this paper is to develop a novel semi-implicit spectral deferred correction (SDC) time marching method. The method can be used in a large class of problems, especially for highly nonlinear ordinary differential equations (ODEs) without easily separating of stiff and non-stiff components, which is more general and efficient comparing with traditional semi-implicit SDC methods. The proposed semi-implicit SDC method is based on low order time integration methods and corrected iteratively. The order of accuracy is increased for each additional iteration. And we also explore its local truncation error analytically. This SDC method is intended to be combined with the method of lines, which provides a flexible framework to develop high order semi-implicit time marching methods for nonlinear partial differential equations (PDEs). In this paper we mainly focus on the applications of the nonlinear PDEs with higher order spatial derivatives, e.g. convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn–Hilliard equation, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the fully discrete schemes are all high order accurate in both space and time, and stable numerically with the time step proportional to the spatial mesh size. Numerical experiments are carried out to illustrate the accuracy and capability of the proposed semi-implicit SDC method. [ABSTRACT FROM AUTHOR]

The miscible displacement of one fluid by another in a porous medium has received considerable attention in subsurface, environmental and petroleum engineering applications. When a fluid of higher mobility displaces another of lower mobility, unstable patterns – referred to as viscous fingering – may arise. Their physical and mathematical study has been the object of numerous investigations over the past century. The objective of this paper is to present a review of these contributions with particular emphasis on variational methods. These algorithms are tailored to real field applications thanks to their advanced features: handling of general complex geometries, robustness in the presence of rough tensor coefficients, low sensitivity to mesh orientation in advection dominated scenarios, and provable convergence with fully unstructured grids. This paper is dedicated to the memory of Dr. Jim Douglas Jr., for his seminal contributions to miscible displacement and variational numerical methods. [ABSTRACT FROM AUTHOR]

In this paper we propose a novel arbitrary high order accurate semi-implicit space–time discontinuous Galerkin method for the solution of the three-dimensional incompressible Navier–Stokes equations on staggered unstructured curved tetrahedral meshes. As is typical for space–time DG schemes, the discrete solution is represented in terms of space–time basis functions. This allows to achieve very high order of accuracy also in time, which is not easy to obtain for the incompressible Navier–Stokes equations. Similarly to staggered finite difference schemes, in our approach the discrete pressure is defined on the primary tetrahedral grid, while the discrete velocity is defined on a face-based staggered dual grid. While staggered meshes are state of the art in classical finite difference schemes for the incompressible Navier–Stokes equations, their use in high order DG schemes is still quite rare. A very simple and efficient Picard iteration is used in order to derive a space–time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. From numerical experiments we find that the linear system seems to be reasonably well conditioned, since all simulations shown in this paper could be run without the use of any preconditioner, even up to very high polynomial degrees. For a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point, the resulting system is, in addition, symmetric and positive definite. This allows us to use even faster iterative solvers, like the conjugate gradient method. The flexibility and accuracy of high order space–time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both, space and time. The proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3D test problems and by comparing the obtained numerical results with available exact analytical solutions, or with other numerical or experimental reference data. To the knowledge of the authors, this is the first time that a space–time discontinuous Galerkin finite element method is presented for the three-dimensional incompressible Navier–Stokes equations on staggered unstructured tetrahedral grids. [ABSTRACT FROM AUTHOR]

In this paper, we design high-order bound-preserving discontinuous Galerkin (DG) methods for multicomponent chemically reacting flows. In this problem, the density and pressure are positive and the mass fractions are between 0 and 1. There are three main difficulties. First of all, it is not easy to construct high-order positivity-preserving schemes for convection-diffusion equations. In this paper, we design a special penalty term to the diffusion term and construct the positivity-preserving flux for the system. The proposed idea is locally conservative, high-order accurate and easy to implement. Secondly, the positivity-preserving technique cannot preserve the upper bound 1 of the mass fractions. To bridge this gap, we apply the positivity-preserving technique to each mass fraction and develop consistent numerical fluxes in the system and conservative time integrations to preserve the summation of the mass fractions to be 1. Therefore, each mass fraction would be between 0 and 1. Finally, most previous bound-preserving DG methods are based on Euler forward time discretization. However, due to the rapid reaction rates, the target system may contain stiff sources, leading to restricted time step sizes. To fix this and preserve conservative property, we apply the conservative modified exponential Runge-Kutta method. The method is third-order accurate and keeps the summation of the mass fractions to be 1. Numerical experiments will be given to demonstrate the good performance of the proposed schemes. • Construct the positivity-preserving fluxes for multicomponent chemically reacting flows. • Apply the conservative modified exponential Runge-Kutta methods. • The technique can be applied to stiff source with large time step. [ABSTRACT FROM AUTHOR]

We propose a generalized Eulerian-Lagrangian (GEL) discontinuous Galerkin (DG) method. The method is a generalization of the Eulerian-Lagrangian (EL) DG method for transport problems proposed in Cai et al. (2021) [5] , which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping size with stability. The newly proposed GEL DG method in this paper is motivated for solving linear hyperbolic systems with variable coefficients, where the velocity field for adjoint problems of the test functions is frozen to constant. In this paper, in a simplified scalar setting, we propose the GEL DG methodology by freezing the velocity field of adjoint problems, and by formulating the semi-discrete scheme over the space-time region partitioned by linear lines approximating characteristics. The fully-discrete schemes are obtained by method-of-lines Runge-Kutta methods. We further design flux limiters for the schemes to satisfy the discrete geometric conservation law (DGCL) and maximum principle preserving (MPP) properties. Numerical results on 1D and 2D linear transport problems are presented to demonstrate great properties of the GEL DG method. These include the high order spatial and temporal accuracy, stability with extra large time stepping size, and satisfaction of DGCL and MPP properties. • We develop a generalized Eulerian Lagrangian (EL) discontinuous Galerkin (DG) method. • We establish connection of the generalized EL DG method with the EL DG method. • Numerical results on linear transport problems. [ABSTRACT FROM AUTHOR]

• Construct implicit pressure explicit concentration methods. • Add a correction step to gain second-order accuracy in time. • The method saves 90% of the CPU time. • The full algorithm preserves the physical bounds of the concentration. In this paper, we construct bound-preserving interior penalty discontinuous Galerkin (IPDG) methods with a second-order implicit pressure explicit concentration (SIPEC) time marching for the coupled system of two-component compressible miscible displacements. The SIPEC method is a crucial innovation based on the traditional second-order strong-stability-preserving Runge-Kutta (SSP-RK2) method. The main idea is to treat the pressure equation implicitly and the concentration equation explicitly. However, this treatment would result in a first-order accurate scheme. Therefore, in all previous works, only the combination of forward and backward Euler time integrations was considered. In this paper, we propose a correction stage to compensate for the second-order accuracy in each time step. There are two main difficulties in constructing a second-order scheme. Firstly, in the concentration equation, correction of the diffusion term will cause anti-diffusion, leading to malfunction of the bound-preserving technique. We can deal with the velocity in the diffusion term explicitly to avoid correction of the diffusion term. Secondly, we need to ensure that the bound-preserving technique for the convection and source terms can be applied when the correction stage has been established. In fact, in the correction stage, the new approximation to the concentration can be chosen as the numerical solution in the previous stage, so the numerical cell averages are positivity-preserving. Moreover, we use the same correction technique for the pressure, so that the consistent flux pairs would guarantee the preservation of the upper bound 1 of the concentration. Numerical experiments will be given to demonstrate that the proposed scheme can reduce the computational cost significantly compared with explicit schemes if the diffusion coefficient D is small in the concentration equation. Moreover, the proposed method also yields much larger cfl number compared with first-order implicit pressure explicit concentration schemes. Moreover, the effectiveness of the bound-preserving technique will also be presented. [ABSTRACT FROM AUTHOR]

The paper is concerned with the numerical solution of the linear steady-state regularized 13-moment equations in two space dimensions. To facilitate the understanding of the specific challenges, the equations are first divided into two subsystems before the full system is approached. The arising problems mainly stem from the complicated saddle-point structure as well as the non-standard nature of the boundary conditions. A continuous interior penalty method is presented and the pronounced advantages of utilizing high order basis functions in this setting are illustrated. To render the presented approach more efficient, a hybridization technique is presented that originates in the discontinuous Galerkin method. [ABSTRACT FROM AUTHOR]

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained. • The Krylov implicit integration factor discontinuous Galerkin methods were first designed on sparse grids. • The new methods can simulate high spatial dimensional reaction-diffusion equations efficiently. • Both theoretical analysis and numerical experiments were performed to study the new methods. [ABSTRACT FROM AUTHOR]

Conservation of energy, spurious-free, and optimal convergence are among the most favorite ingredients of a discontinuous Galerkin finite-element time-domain method (DG-FETD) for the analysis of transient electromagnetic problems. Unfortunately, due to the reliance on dissipative upwind flux to suppress spurious modes, most existing DG-FETDs possess only one or two of them. In this paper, we present a novel two-dimensional DG-FETD method, which combines all of the three advantages. The proposed method is based on the electric field E and magnetic field H , whose approximation spaces are composed of Whitney's edge functions. To inhibit the spurious modes while keeping the conservation of numerical energy, the numerical flux with dissipation is not adopted. Instead, a carefully designed interface condition along with the non-dissipative central flux is employed for this purpose. That is, across each face of an element except for those coinciding with boundaries of the computational domain, if E is strongly enforced to be tangential continuous then the continuity of H is weakly imposed, and vice versa. For the convenience of complying with this requirement and without loss of the inherent parallelism of the DG-FETD method, two partially staggered sets of subdomains are built for E and H , respectively. These subdomains are made up of sub-elements generated through a refinement of each cell on the initial mesh into four similar elements. The final discrete system is advanced with a second order leap-frog time-stepping scheme. A series of numerical examples demonstrate that the present method is robust and much more superior to the conventional DG-FETD methods. [ABSTRACT FROM AUTHOR]

Abstract In most classical approaches of computational geophysics for seismic wave propagation problems, complex surface topography is either accounted for by boundary-fitted unstructured meshes, or, where possible, by mapping the complex computational domain from physical space to a topologically simple domain in a reference coordinate system. However, all these conventional approaches face problems if the geometry of the problem becomes sufficiently complex. They either need a mesh generator to create unstructured boundary-fitted grids, which can become quite difficult and may require a lot of manual user interactions in order to obtain a high quality mesh, or they need the explicit computation of an appropriate mapping function from physical to reference coordinates. For sufficiently complex geometries such mappings may either not exist or their Jacobian could become close to singular. Furthermore, in both conventional approaches low quality grids will always lead to very small time steps due to the Courant-Friedrichs-Lewy (CFL) condition for explicit schemes. In this paper, we propose a completely different strategy that follows the ideas of the successful family of high resolution shock-capturing schemes, where discontinuities can actually be resolved anywhere on the grid, without having to fit them exactly. We address the problem of geometrically complex free surface boundary conditions for seismic wave propagation problems with a novel diffuse interface method (DIM) on adaptive Cartesian meshes (AMR) that consists in the introduction of a characteristic function 0 ≤ α ≤ 1 which identifies the location of the solid medium and the surrounding air (or vacuum) and thus implicitly defines the location of the free surface boundary. Physically, α represents the volume fraction of the solid medium present in a control volume. Our new approach completely avoids the problem of mesh generation, since all that is needed for the definition of the complex surface topography is to set a scalar color function to unity inside the regions covered by the solid and to zero outside. The governing equations are derived from ideas typically used in the mathematical description of compressible multiphase flows. An analysis of the eigenvalues of the PDE system shows that the complexity of the geometry has no influence on the admissible time step size due to the CFL condition. The model reduces to the classical linear elasticity equations inside the solid medium where the gradients of α are zero, while in the diffuse interface zone at the free surface boundary the governing PDE system becomes nonlinear. We can prove that the solution of the Riemann problem with arbitrary data and a jump in α from unity to zero yields a Godunov-state at the interface that satisfies the free-surface boundary condition exactly, i.e. the normal stress components vanish. In the general case of an interface that is not aligned with the grid and which is not infinitely thin, a systematic study on the distribution of the volume fraction function inside the interface and the sensitivity with respect to the thickness of the diffuse interface layer has been carried out. In order to reduce numerical dissipation, we use high order discontinuous Galerkin (DG) finite element schemes on adaptive AMR grids together with a second order accurate high resolution shock capturing subcell finite volume (FV) limiter in the diffuse interface region. We furthermore employ a little dissipative HLLEM Riemann solver, which is able to resolve the steady contact discontinuity associated with the volume fraction function and the spatially variable material parameters exactly. While the method is locally high order accurate in the regions without limiter, the global order of accuracy of the scheme is at most two if the limiter is activated. It is locally of order one inside the diffuse interface region, which is typical for shock-capturing schemes at shocks and contact discontinuities. We show a large set of computational results in two and three space dimensions involving complex geometries where the physical interface is not aligned with the grid or where it is even moving. For all test cases we provide a quantitative comparison with classical approaches based on boundary-fitted unstructured meshes. Highlights • First application of a diffuse interface approach to linear elasticity in 2D and 3D. • Even complex physical geometries are simply represented by a scalar volume fraction function. • The new method allows to avoid the mesh generation problem in computational seismology. • No problems with low-quality meshes, sliver elements and the associated small time steps any more. • Fully automated workflow for the computational setup of the problem via AMR and subcell limiter. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a Runge-Kutta Discontinuous Galerkin (RKDG) method is proposed for solving the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations under the Lagrangian framework. The method is based on the Lagrangian type scheme designed for the Euler equations of compressible gas dynamics on arbitrary quadrilateral meshes. To deal with the divergence free constraint condition of the magnetic field, we adopt the local divergence free function space to approximate the magnetic field. A HWENO reconstruction limiter is presented for the numerical solution to inhibit the non-physical oscillation for flows with strong discontinuities. This limiter can not only preserve the local divergence free property of the magnetic field but also avoid to calculate the complex eigen-system of the MHD equations. Some numerical examples are presented to demonstrate the accuracy and the non-oscillatory property of the scheme. Highlights • A Lagrangian method to solve the 2D MHD equations on arbitrary quadrilateral meshes. • The local divergence free function space is used to approximate the magnetic field. • The limiter can preserve the local divergence free property of the magnetic field. [ABSTRACT FROM AUTHOR]

Abstract Constructing stable difference schemes on complex geometries is an arduous task. Even fairly simple partial differential equations end up very convoluted in their discretized form, making them difficult to implement and manage. Spatial discretizations using so called summation-by-parts operators have mitigated this issue to some extent, particularly on rectangular domains, making it possible to formulate stable discretizations in a compact and understandable manner. However, the simplicity of these formulations is lost for curvilinear grids, where the standard procedure is to transform the grid to a rectangular one, and change the structure of the original equation. In this paper we reinterpret the grid transformation as a transformation of the summation-by-parts operators. This results in operators acting directly on the curvilinear grid. Together with previous developments in the field of nonconforming grid couplings we can formulate simple, implementable, and provably stable schemes on general nonconforming curvilinear grids. The theory is applicable to methods on summation-by-parts form, including finite differences, discontinuous Galerkin spectral element, finite volume, and flux reconstruction methods. Time dependent advection–diffusion simulations corroborate the theoretical development. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we construct an efficient numerical scheme for full-potential electronic structure calculations of periodic systems. In this scheme, the computational domain is decomposed into a set of atomic spheres and an interstitial region, and different basis functions are used in different regions: radial basis functions times spherical harmonics in the atomic spheres and plane waves in the interstitial region. These parts are then patched together by discontinuous Galerkin (DG) method. Our scheme has the same philosophy as the widely used (L)APW methods in materials science, but possesses systematically spectral convergence rate. We provide a rigorous a priori error analysis of the DG approximations for the linear eigenvalue problems, and present some numerical simulations in electronic structure calculations. [ABSTRACT FROM AUTHOR]

Abstract Second and higher order numerical approximations of conservation laws for vector fields call for the use of limiting techniques based on generalized monotonicity criteria. In this paper, we introduce a family of directional vertex-based slope limiters for tensor-valued gradients of formally second-order accurate piecewise-linear discontinuous Galerkin (DG) discretizations. The proposed methodology enforces local maximum principles for scalar products corresponding to projections of a vector field onto the unit vectors of a frame-invariant orthogonal basis. In particular, we consider anisotropic limiters based on singular value decompositions and the Gram-Schmidt orthogonalization procedure. The proposed extension to hyperbolic systems features a sequential limiting strategy and a global invariant domain fix. The pros and cons of different approaches to vector limiting are illustrated by the results of numerical studies for the two-dimensional shallow water equations and for the Euler equations of gas dynamics. Highlights • Frame-invariant vector limiters for piecewise-linear vector fields. • Several approaches to definition of objective limiting directions. • Accuracy-preserving sequential limiters for conserved products. • Invariant domain fix for enforcing global positivity constraints. • Quantification of changes caused by rotations of the domain. [ABSTRACT FROM AUTHOR]

Abstract We present in this paper a novel numerical method for the simulation of time-dependent, diluted spray combustion problems. Spray combustion is a complex numerical problem owing to the diversity of the phenomena (convection, diffusion, heat transfer, vaporization of droplets and chemical reaction) and the disparity of the scales involved. Our model considers a large number of tiny droplets (liquid phase) surrounded by a gas flow (gas phase), making up a special two-phase system where droplet interaction is neglected on account of the much larger inter-droplet distance compared to the droplet radius typically found in burners. The mathematical formulation of the problem follows a hybrid, Eulerian–Lagrangian approach: the continuous, gas phase is ruled by the conservation equations of mass, momentum, species mass fractions and energy, while the liquid phase is described by a set of ordinary differential equations (ODE) that govern the properties of each droplet along its fluid trajectory. For the resolution of the conservation equations of the gas phase, a Lagrange–Galerkin method in an anisotropic adaptive finite element framework is used. In combination with the aforementioned Lagrange–Galerkin methodology we apply a second order Explicit Runge–Kutta–Chebyshev scheme, also useful to solve the ODE systems of equations for the droplets movement. Explicit Runge–Kutta–Chebyshev preserves numerical stability while also facilitating a decoupled calculation of the unknown variables of the gas and liquid phases, a most beneficial feature from a computationally point of view. Finally, we show several examples to highlight the capabilities of our numerical algorithm in canonical and real-world configurations. Highlights • Eulerian–Lagrangian formulation for time-dependent, spray combustion problems. • Lagrange–Galerkin numerical scheme in anisotropic adaptive finite element for gas phase. • Lagrange–Galerkin scheme decouples Navier–Stokes from the energy/species mass fraction conservation equations. • Ex-RKC tackles coupled, highly-stiff, non-linear systems with enhanced numerical stability. • Simulations of canonical and real-world spray combustion configurations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a third-order reconstructed DG(p 1 p 2) method based on the variational reconstruction (VR) (Wang et al., 2017 [24]) is developed for simulating the two dimensional steady-state compressible flows on unstructured and hybrid grids. The proposed method combines the advantages of the DG discretization with the flexibility of the variational reconstruction, which exhibits its superior potential in enhancing the level of accuracy compared to the underlying DG method. In this variational rDG(p 1 p 2) method, the low order degrees of freedom are evolved through the underlying DG(p 1) method, while the high order degrees of freedom are reconstructed through the variational reconstruction, in which the constitutive relations are built by minimizing the so-called 'cost function'. The cost function is defined by the total interfacial jump integration in the computational domain using the variational method. The large sparse linear system resulted by the variational reconstruction is solved in an efficient way coupled with the temporal discretization for the steady-state simulations. A number of test cases are presented to assess the performance of the new high order variational rDG(p 1 p 2) method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, an efficient, high-order accurate, level set reinitialisation method is proposed, based on the elliptic reinitialisation method (Basting and Kuzmin, 2013 [1]), which is discretised spatially using the discontinuous Galerkin (DG) symmetric interior penalty method (SIPG). In order to achieve this a number of improvements have been made to the elliptic reinitialisation method including; reformulation of the underlying minimisation problem driving the solution; adoption of a Lagrange multiplier approach for enforcing a Dirichlet boundary condition on the implicit level set interface; and adoption of a narrow band approach. Numerical examples confirm the high-order accuracy of the resultant method by demonstrating experimental orders of convergence congruent with optimal convergence rates for the SIPG method, that is h p + 1 and h p in the L 2 and DG norms respectively. Furthermore, the degree to which the level set function satisfies the Eikonal equation improves proportionally to h p , and the often ignored homogeneous Dirichlet boundary condition on the interface is shown to be satisfied accurately with a rate of convergence of at least h 2 for all polynomial orders. [ABSTRACT FROM AUTHOR]

Abstract A high-order numerical method for interface-preserving level-set reinitialization is presented in this paper. In the interface cells, the gradient of the level-set function is determined by a weighted local projection scheme and the missing additive constant is determined such that the position of the zero level set is preserved. In the non-interface cells, we compute the gradient of the level-set function by solving a Hamilton–Jacobi equation as a conservation law system using the discontinuous Galerkin method, following the work by Hu and Shu [SIAM J. Sci. Comput. 21 (1999) 660–690]; the missing constant is then recovered by the continuity of the level-set function while taking into account the characteristics. To handle highly distorted initial conditions, we develop a hybrid numerical flux that combines the Lax–Friedrichs flux and the penalty flux. Our method is stable for non-trivial test cases and handles singularities away from the interface very well. When derivative singularities are present on the interface, a second-derivative limiter is designed to suppress the oscillations. At least (N + 1) th order accuracy in the interface cells and N th order in the whole domain are observed for smooth solutions when N th degree polynomials are used. Two dimensional test cases are presented to demonstrate superior properties such as accuracy, long-term stability, interface-preserving capability, and easy treatment of contact lines. We also show some preliminary results on the pinch-off process of a pendant drop, where topological changes of the fluid interface are involved. Our method is readily extendable to three dimensions and adaptive meshes. [ABSTRACT FROM AUTHOR]

Highlights • Combining multigrid and p-adaptation methods speeds up convergence significantly. • Anisotropic τ -estimation can be performed cheaply in an anisotropic multigrid cycle. • Multi-stage p-adaptation methods based on full multigrid improve performance. Abstract High-order discontinuous Galerkin methods have become a popular technique in computational fluid dynamics because their accuracy increases spectrally in smooth solutions with the order of the approximation. However, their main drawback is that increasing the order also increases the computational cost. Several techniques have been introduced in the past to reduce this cost. On the one hand, local mesh adaptation strategies based on error estimation have been proposed to reduce the number of degrees of freedom while keeping a similar accuracy. On the other hand, multigrid solvers may accelerate time marching computations for a fixed number of degrees of freedom. In this paper, we combine both methods and present a novel anisotropic p-adaptation multigrid algorithm for steady-state problems that uses the multigrid scheme both as a solver and as an anisotropic error estimator. To achieve this, we show that a recently developed anisotropic truncation error estimator [1, A. M. Rueda-Ramírez, G. Rubio, E. Ferrer, E. Valero, Truncation error estimation in the p-anisotropic discontinuous Galerkin spectral element method, J. Sci. Comput.] is perfectly suited to be performed inside the multigrid cycle with negligible extra cost. Furthermore, we introduce a multi-stage p-adaptation procedure which reduces the computational time when very accurate results are required. The proposed methods are tested for the compressible Navier–Stokes equations, where we investigate two steady-state cases. First, the 2D boundary layer flow on a flat plate is studied to assess accuracy and computational cost of the algorithm, where a speed-up of 816 is achieved compared to the traditional explicit method. Second, the 3D flow around a sphere is simulated and used to test the anisotropic properties of the proposed method, where a speed-up of 152 is achieved compared to the explicit method. The proposed multi-stage procedure achieved a speed-up of 2.6 in comparison to the single-stage method in highly accurate simulations. [ABSTRACT FROM AUTHOR]

Abstract Local discontinuous Galerkin (LDG) methods are popular for convection-diffusion equations. In LDG methods, we introduce an auxiliary variable p to represent the derivative of the primary variable u , and solve them on the same mesh. It is well known that the maximum-principle-preserving (MPP) LDG method is only available up to second-order accuracy. Recently, we introduced a new algorithm, and solve u and p on different meshes, and obtained stability and optimal error estimates. In this paper, we will continue this approach and construct MPP third-order LDG methods for convection-diffusion equations on overlapping meshes. The new algorithm is more flexible and does not increase any computational cost. Numerical evidence will be given to demonstrate the accuracy and good performance of the third-order MPP LDG method. Highlights • Introduce a new LDG method on overlapping meshes. • Construct third-order maximum-principle-preserving LDG scheme on the overlapping meshes. • A special technique is applied to extend the idea to nonlinear equations. [ABSTRACT FROM AUTHOR]

Abstract The purpose of the present paper is to compare two semi-Lagrangian methods in the context of the four-dimensional Vlasov–Poisson equation. More specifically, our goal is to compare the performance of the more recently developed semi-Lagrangian discontinuous Galerkin scheme to cubic spline interpolation, which is widely used in Eulerian Vlasov simulation. To that end, we perform simulations for nonlinear Landau damping and a two-stream instability and provide benchmarks for the SeLaLib and SLDG codes, both on a workstation and using MPI on a cluster. In addition, we will present results for the graphic processing unit (GPU) implementation contained in SLDG. We find that the semi-Lagrangian discontinuous Galerkin scheme shows a moderate improvement in run time for nonlinear Landau damping and a substantial improvement for the two-stream instability. It should be emphasized that these results are markedly different from results obtained in the asymptotic regime, which favor spline interpolation. Thus, we conclude that the traditional approach of evaluating numerical methods is misleading, even for short time simulations. In addition, the absence of any global communication in the semi-Lagrangian discontinuous Galerkin method gives it a decisive advantage for scaling to more than 256 cores. Highlights • The SLDG method is efficient for high dimensional Vlasov simulation. • We obtain superior performance compared to a well established implementation. • The local nature of SLDG helps significantly in parallelization (e.g. on GPUs). • The performance in the asymptotic regime is a very poor predictor of accuracy. • First performance comparison of SLDG in more than two dimensions. [ABSTRACT FROM AUTHOR]

Abstract The paper proposes a novel computational method for simulating diffraction of obliquely incident plane waves by a doubly periodic array of arbitrarily shaped planar delaminations in elastic layered structures. The presented approach is an extension of the boundary integral equation method (BIEM) based on the boundary integral approach. Periodically situated delaminations are simulated via spring boundary conditions (SBCs), which allows considering bridged cracks. A boundary integral equation is formulated for a reference delamination based on SBCs in terms of the Fourier transform of Green's matrix and solved using Galerkin method. Two modifications of the method are proposed: the spectral BIEM applied only for rectangular delaminations and the meshless BIEM with axisymmetric basis functions, which is applied for delaminations of arbitrary shapes. The effectiveness of regular and irregular meshes is studied for the meshless BIEM. Comparison of both modifications of the BIEM is provided for open and bridged cracks. The proposed method allows considering specific wave phenomena such as a band-gap formation or opening of an interface for wave propagation in elastic layered structures with a doubly periodic array of inhomogeneities (e.g. phononic crystals and damaged interfaces). Highlights • Two modifications of the boundary integral equation method are proposed. • The method simulates wave diffraction by a doubly periodic array of delaminations. • Convergence is shown for open and bridged cracks of different shapes. • The developed method is an effective tool for studying specific wave phenomena. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the bipolar Boltzmann-Poisson model, both for the deterministic system and the system with uncertainties, with asymptotic behavior leading to the drift diffusion-Poisson system as the Knudsen number goes to zero. The random inputs can arise from collision kernels, doping profile and initial data. We adopt a generalized polynomial chaos approach based stochastic Galerkin (gPC-SG) method. Sensitivity analysis is conducted using hypocoercivity theory for both the analytical solution and the gPC solution for a simpler model that ignores the electric field, and it gives their convergence toward the global Maxwellian exponentially in time. A formal proof of the stochastic asymptotic-preserving (s-AP) property and a uniform spectral convergence with error decaying exponentially in time in the random space of the scheme is given. Numerical experiments are conducted to validate the accuracy, efficiency and asymptotic properties of the proposed method. Highlights • The bipolar Boltzmann–Poisson model with random inputs and diffusive scaling. • Asymptotic limit of the model leads to the drift diffusion-Poisson system. • Sensitivity analysis for both the analytic and gPC solution. • A uniform spectral convergence of the gPC-SG scheme with the gPC error decaying exponentially in time. • The gPC-SG scheme is stochastic asymptotic-preserving. [ABSTRACT FROM AUTHOR]

Abstract The time-dependent equations of computational electrodynamics (CED) are evolved consistent with the divergence constraints on the electric displacement and magnetic induction vector fields. Respecting these constraints has proved to be very useful in the classic finite-difference time-domain (FDTD) schemes. As a result, there has been a recent effort to design finite volume time domain (FVTD) and discontinuous Galerkin time domain (DGTD) schemes that satisfy the same constraints and, nevertheless, draw on recent advances in higher order Godunov methods. This paper catalogues the first step in the design of globally constraint-preserving DGTD schemes. The algorithms presented here are based on a novel DG-like method that is applied to a Yee-type staggering of the electromagnetic field variables in the faces of the mesh. The other two novel building blocks of the method include constraint-preserving reconstruction of the electromagnetic fields and multidimensional Riemann solvers; both of which have been developed in recent years by the first author. The resulting DGTD scheme is linear, at least when limiters are not applied to the DG scheme. As a result, it is possible to carry out a von Neumann stability analysis of the entire suite of DGTD schemes for CED at orders of accuracy ranging from second to fourth. The analysis requires some simplifications in order to make it analytically tractable, however, it proves to be extremely instructive. A von Neumann stability analysis is a necessary precursor to the design of a full DGTD scheme for CED. It gives us the maximal CFL numbers that can be sustained by the DGTD schemes presented here at all orders. It also enables us to understand the wave propagation characteristics of the schemes in various directions on a Cartesian mesh. We find that constraint-preserving DGTD schemes permit CFL numbers that are competitive with conventional DG schemes. However, like conventional DG schemes, the CFL of DGTD schemes decreases with increasing order. To counteract that, we also present constraint-preserving PNPM schemes for CED. We find that the third and fourth order constraint-preserving DGTD and P1PM schemes have some extremely attractive properties when it comes to low-dispersion, low-dissipation propagation of electromagnetic waves in multidimensions. Numerical accuracy tests are also provided to support the von Neumann stability analysis. We expect these methods to play a role in those problems of engineering CED where exceptional precision must be achieved at any cost. Highlights • Globally constraint-preserving DGTD and PMPM schemes for CED. • DG-like method with facial collocation of field variables. • Constraint-preserving reconstruction and multidimensional Riemann Solvers for CED. • von Neumann stability analysis provides insights on CFL and wave propagation. • The P1PN schemes are competitive with DGTD schemes, but permit much larger CFL. [ABSTRACT FROM AUTHOR]

Abstract This paper presents an analysis of refinement indicators for the simulation of steady and unsteady flows by means of p -adaptive algorithms for discontinuous Galerkin (DG) methods. Residual-error, discretization-error and feature-based indicators are compared by studying their effect on convergence history, computational gain and refinement regions selected by the adaptive algorithm. The analysis is initially carried out on steady flow configurations. Static p -adaptive simulations of the periodic flow past a cylinder at R e = 100 are then performed. Compared to uniform p -refinement, a reduction between 50% and 75% in the number of degrees of freedom is obtained for all test cases considered. The accuracy and efficiency observed for the small-scale energy density [1] and spectral decay [2] indicators demonstrate their great potential for the adaptive simulation of unsteady flows. Highlights • Discretization and residual error refinement indicators for DG methods are analysed. • p -adaptive simulations of steady and unsteady configurations are performed. • Performance is compared based on convergence history and refinement distribution. • Computational cost and possible implementation issues are discussed. [ABSTRACT FROM AUTHOR]