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We present an hp-adaptive discretization for a sharp interface model with a level-set ghost-fluid method to simulate compressible multiphase flows. The scheme applies an efficient p-adaptive discontinuous Galerkin (DG) operator in regions of smooth flow. Shocks and the phase interface are captured by a Finite Volume (FV) scheme on a h-refined element-local sub-grid. The resulting hp-adaptive scheme thus combines both the high order accuracy of the DG method and the robustness of the FV scheme by using p-adaptation in smooth areas and h-refinement at discontinuities, respectively. For the level-set based interface tracking, a similar hybrid DG/FV operator is employed. Both p-refinement and FV shock and interface capturing are performed at runtime and controlled by an indicator, which is based on the modal decay of the solution polynomials. In parallel simulations, the hp-adaptive discretization together with the costly interface tracking algorithm cause a significant imbalance in the processor workloads. To ensure parallel efficiency, we propose a dynamic load balancing scheme that determines the workload distribution by element-local wall time measurements and redistributes elements along a space filling curve. The parallelization strategy is supported by strong scaling tests using up to 8192 cores. The framework is applied to established benchmarks problems for inviscid, compressible multiphase flows. The results demonstrate that the hybrid adaptive discretization can efficiently and accurately handle complex multiphase flow problems involving pronounced interface deformations and merging interface contours. [ABSTRACT FROM AUTHOR]

Hypoxy induced angiogenesis processes can be described by coupling an integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the angiogenic factor. We propose high order positivity preserving schemes to approximate the marginal tip density by combining an asymptotic reduction with weighted essentially non oscillatory and strong stability preserving time discretization. We capture soliton-like solutions representing blood vessel formation and spread towards hypoxic regions. [ABSTRACT FROM AUTHOR]

The classical paper [40] by S. K. Godunov had a revolutionary effect on the field of numerical simulations of compressible fluid flows. Yet, for some twenty years after its publication, it was mostly used by the mathematical community in the Soviet Union. The seminal paper of van Leer [85] has inaugurated the period of universal interest in high-resolution extensions of Godunov's scheme. The first step to take was to modify the (locally) self-similar solution to the Riemann Problem (at discontinuities) by allowing piecewise-polynomial (rather than piecewise-constant) initial data. The GRP (Generalized Riemann Problem) analysis [6] provided analytical solutions (for piecewise-linear data) that could be readily implemented in a high-resolution robust code. This review paper focuses on the evolution of some mathematical aspects of this method. It addresses the three concepts of consistency, stability and convergence in the context of compact finite difference (or finite volume) schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of "balance laws", a common viewpoint in relevant physical conservation laws. The first significant observation is that under very mild conditions a weak solution is indeed a solution to the balance law (obtained by a formal application of the Gauss-Green formula). Since high-resolution schemes require the computation of several quantities per mesh cell (e.g., slopes), the notion of "consistency" must be extended to this framework. A combination of consistency hypothesis with stability of the scheme leads to a suitable convergence theorem, generalizing the classical convergence theorem of Lax and Wendroff [52]. Finally, the limit functions are shown to be entropy solutions by using a notion of "Godunov compatibility", which serves as a substitute to the entropy condition. • The concept of "consistency" is explored in detail, for general high resolution schemes. • Consistency (and its order) of a finite difference scheme depends on the underlying space of approximation. • A generalization of the Lax-Wendroff convergence theorem is given. • The "entropy condition" is replaced by the concept of "Godunov compatibility". [ABSTRACT FROM AUTHOR]

• CFD simulation of heat convective-diffusive solidification of non-Newtonian fluid. • Fast-accurate prediction of heat transfer in phase change by SIMPLERnP/FVM scheme. • Ternary alloy solidification at high Rayleigh number with Herschel-Bulkley rheology. • Variations of power index and yield stress change leading heat transfer mechanism. • Temperature variation of liquid phase change fraction a key issue in ternary alloys. Numerical modeling of convective heat transfer employing non-Newtonian fluids is a key issue and represent an interesting challenge in many engineering applications. For this reason, this work develops an enhanced numerical model to predict fast and accurately the convective phase change for generalized non-Newtonian fluids. The strongly thermally coupled model is solved by an improved pressure-correction algorithm (SIMPLERnP) and the Finite Volume Method with accuracy of the third order for the transient terms and second order for the convective boundary conditions and the velocity gradients that allows to calculate the apparent viscosity. The study examines the natural heat convective solidification of the ternary Al-27 %Cu-5.25 %Si alloy with the Herschel-Bulkley rheology at a high Rayleigh number (>107). In addressing the main problem, four crucial aspects are discussed: the validation of the numerical scheme with numerical and experimental results; the modeling of the non-linear temperature variation of the liquid phase change fraction; the non-Newtonian effects of the power index (0.1 ≤ n ≤ 1.9) and yield stress (0≤ τ 0 ≤5 Pa) on the heat transfer mechanisms; and finally, the time of computation with the high-order numerical scheme and the enhanced pressure-correction algorithm. The main finding is that FVM/SIMPLERnP provides a significant reduction in the time of computation compared to reliable pressure-correction schemes (SIMPLER and IDEAL), being a robust, accurate and efficient scheme to solve complex natural heat convective phase change problems involve generalized non-Newtonian fluids at high-Rayleigh numbers. [ABSTRACT FROM AUTHOR]

In this work, we present a novel hybrid Discontinuous Galerkin scheme with hp-adaptivity capabilities for the compressible Euler equations. In smooth regions, an efficient and accurate discretization is achieved via local p-adaptation. At strong discontinuities and shocks, a finite volume scheme on an h-refined element-local subgrid gives robustness. Thus, we obtain a hp-adaptive scheme that exploits both the high convergence rate and efficiency of a p-adaptive high order scheme as well as the stable and accurate shock capturing abilities of a low order finite volume scheme, but avoids the inherent resolution loss through h-refinement. A single a priori indicator, based on the modal decay of the local polynomial solution representation, is used to distinguish between discontinuous and smooth regions and control the p-refinement. Our method is implemented as an extension to the open source software FLEXI. Hence, the efficient implementation of the method for high performance computers was an important criterion during the development. The efficiency of our adaptive scheme is demonstrated for a variety of test cases, where results are compared against non adaptive simulations. Our findings suggest that the proposed adaptive method produces comparable or even better results with significantly less computational costs. [ABSTRACT FROM AUTHOR]

New criteria for energy stability of multi-step, multi-stage, and mixed schemes are introduced in the context of evolution equations that arise as gradient flow with respect to a metric. These criteria are used to exhibit second and third order consistent, energy stable schemes, which are then demonstrated on several partial differential equations that arise as gradient flow with respect to the 2-Wasserstein metric. • Energy stability conditions for multi-step, multi-stage, and mixed schemes for gradient flow in arbitrary metric spaces. • New criteria are proposed based on embedding of finite graphs into Euclidean spaces. • Second order energy stable and third order energy bounded schemes are exhibited. [ABSTRACT FROM AUTHOR]

Recently, the interest in the study of various types of transients involving changes in the boron concentration inside the reactor, has led to an increase in the interest of developing and studying new models and tools that allow a correct study of boron transport. Therefore, a significant variety of different boron transport models and spatial difference schemes are available in the thermal-hydraulic codes. According to this interest, in this work it will be compared the results obtained using the different boron transport models implemented in the NRC thermal-hydraulic code TRACE. To do this, a set of models has been created using the different options and configurations that could have influence in boron transport. These models allow us to reproduce a simple event of filling or emptying the boron concentration in a long pipe. Moreover, with the aim to compare the differences obtained when one-dimensional or three-dimensional components are chosen, it has modeled many different cases using only pipe components or a mix of pipe and vessel components. In addition, the influence of the void fraction in the boron transport has been studied and compared under close conditions to a BWR commercial model. A final collection of the different cases and boron transport models are compared between them and those corresponding to the analytical solution provided by the Burgers equation. From this comparison, important conclusions are drawn that will be the basis of modeling the boron transport in TRACE adequately.

In the present paper, a multi-step reconstruction procedure is proposed for high order finite volume schemes on unstructured grids using compact stencil. The procedure is a recursive algorithm that can eventually provide sufficient relations for high order reconstruction in a multi-step procedure. Two key elements of this procedure are the partial inversion technique and the continuation technique. The partial inversion can be used not only to obtain lower order reconstruction based on existing reconstruction relations, but also to regularize the existing reconstruction relations to provide new relations for higher order reconstructions. The continuation technique is to extend the regularized relations on the face-neighboring cells to current cell as additional reconstruction relations. This multi-step procedure is operationally compact since in each step only the relations defined on a compact stencil are used. In the present paper, the third and fourth order finite volume schemes based on two-step quadratic and three-step cubic reconstructions are studied. [ABSTRACT FROM AUTHOR]

Abstract A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P 4 T 2 − BVD (polynomial of 4-degree and THINC function of 2-level reconstruction based on BVD algorithm), is formulated as a two-stage spatial reconstruction scheme following the BVD (Boundary Variation Diminishing) principle that minimizes the jumps of the reconstructed values at cell boundaries. In the P 4 T 2 − BVD scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) functions with two-level steepness are used as the candidate reconstruction functions. The final reconstruction function is selected through the two-stage BVD algorithm so as to effectively control both numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the P 4 T 2 − BVD scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it produces accurate solutions in long term computation. P 4 T 2 − BVD , as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales. Highlights • An innovative fifth-order shock capturing scheme is proposed. • Low-dissipation linear scheme is retrieved for smooth solution over all wave numbers. • Discontinuous and smooth solution of wide-band scales are simultaneously solved with high accuracy. • The free-mode solutions are faithfully maintained in long term computation. [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]

This paper aims to introduce a new class of high order conservative schemes to solve systems of conservation laws. The idea is to couple the conservation form of the system with, possibly simpler, alternative formulations, which can be used to speed up the time update. In this work, we illustrate the procedure for a Runge{Kutta time advancement, but other choices are possible. We show that, as long as the last update is carried out in conservative form, all internal stages can be computed using any consistent nonconservative formulation, still ensuring the propagation of shock waves with the correct speeds. The same procedure can be easily extended to finite difference schemes. Tests from classical and relativistic gas dynamics are carried out to study convergence, numerical robustness and performance. [ABSTRACT FROM AUTHOR]

In this paper we propose a new high order accurate space–time discontinuous Galerkin (DG) finite element scheme for the solution of the linear elastic wave equations in first order velocity-stress formulation in two and three-space dimensions on staggered unstructured triangular and tetrahedral meshes. The method reaches arbitrary high order of accuracy in both space and time via the use of space–time basis and test functions. Within the staggered mesh formulation, we define the discrete velocity field in the control volumes of a primary mesh, while the discrete stress tensor is defined on a face-based staggered dual mesh. The space–time DG formulation leads to an implicit scheme that requires the solution of a linear system for the unknown degrees of freedom at the new time level. The number of unknowns is reduced at the aid of the Schur complement, so that in the end only a linear system for the degrees of freedom of the velocity field needs to be solved, rather than a system that involves both stress and velocity. Thanks to the use of a spatially staggered mesh, the stencil of the final velocity system involves only the element and its direct neighbors and the linear system can be efficiently solved via matrix-free iterative methods. Despite the necessity to solve a linear system, the numerical scheme is still computationally efficient. The chosen discretization and the linear nature of the governing PDE system lead to an unconditionally stable scheme, which allows large time steps even for low quality meshes that contain so-called sliver elements. The fully discrete staggered space–time DG method is proven to be energy stable for any order of accuracy, for any mesh and for any time step size. For the particular case of a simple Crank–Nicolson time discretization and homogeneous material, the final velocity system can be proven to be symmetric and positive definite and in this case the scheme is also exactly energy preserving . The new scheme is applied to several test problems in two and three space dimensions, providing also a comparison with high order explicit ADER-DG schemes. [ABSTRACT FROM AUTHOR]

This paper presents an efficient procedure for overcoming the deficiency of weighted essentially non-oscillatory schemes near discontinuities. Through a thorough incorporation of smoothness indicators into the weights definition, up to ninth-order accurate multistep methods are devised, providing weighted essentially non-oscillatory schemes with enhanced order of convergence at transition points from smooth regions to a discontinuity, while maintaining stability and the essentially non-oscillatory behavior. We also provide a detailed analysis of the resolution power and show that the solution enhancements of the new method at smooth regions come from their ability to render smoothness indicators closer to uniformity. The new scheme exhibits similar fidelity as other multistep schemes; however, with superior characteristics in terms of robustness and efficiency, as no logical statements or mapping function is needed. Extensions to higher orders of accuracy present no extra complexity. Numerical solutions of linear advection problems and nonlinear hyperbolic conservation laws are used to demonstrate the scheme's improved behavior for shock-capturing problems. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We establish a unified framework for L² error analysis for high order Lagrange finite volume methods on triangular meshes. Orthogonal conditions are originally proposed to construct dual partitions on triangular meshes, such that the corresponding finite volume method (FVM) schemes hold optimal L² norm convergence order. Moreover, with the Aubin--Nitsche technique, we prove the optimal L² error estimate for high order FVM schemes on triangular meshes. Some numerical experiments are presented to demonstrate the proved result. [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]

We present a second order residual distribution scheme for scalar transport problems in shallow water flows. The scheme, suitable for the unsteady cases, is obtained adapting to the shallow water context the explicit Runge–Kutta schemes for scalar equations [1] . The resulting scheme is decoupled from the hydrodynamics yet the continuity equation has to be considered in order to respect some important numerical properties at discrete level. Beyond the classical characteristics of the residual formulation presented in [1,2] , we introduce the possibility to iterate the corrector step in order to improve the accuracy of the scheme. Another novelty is that the scheme is based on a precise monotonicity condition which guarantees the respect of the maximum principle. We thus end up with a scheme which is mass conservative, second order accurate and monotone. These properties are checked in the numerical tests, where the proposed approach is also compared to some finite volume schemes on unstructured grids. The results obtained show the interest in adopting the predictor–corrector scheme for pollutant transport applications, where conservation of the mass, monotonicity and accuracy are the most relevant concerns. [ABSTRACT FROM AUTHOR]

In various astrophysics settings it is common to have a two-fluid relativistic plasma that interacts with the electromagnetic field. While it is common to ignore the displacement current in the ideal, classical magnetohydrodynamic limit, when the flows become relativistic this approximation is less than absolutely well-justified. In such a situation, it is more natural to consider a positively charged fluid made up of positrons or protons interacting with a negatively charged fluid made up of electrons. The two fluids interact collectively with the full set of Maxwell's equations. As a result, a solution strategy for that coupled system of equations is sought and found here. Our strategy extends to higher orders, providing increasing accuracy. The primary variables in the Maxwell solver are taken to be the facially-collocated components of the electric and magnetic fields. Consistent with such a collocation, three important innovations are reported here. The first two pertain to the Maxwell solver. In our first innovation, the magnetic field within each zone is reconstructed in a divergence-free fashion while the electric field within each zone is reconstructed in a form that is consistent with Gauss' law. In our second innovation, a multidimensionally upwinded strategy is presented which ensures that the magnetic field can be updated via a discrete interpretation of Faraday's law and the electric field can be updated via a discrete interpretation of the generalized Ampere's law. This multidimensional upwinding is achieved via a multidimensional Riemann solver. The multidimensional Riemann solver automatically provides edge-centered electric field components for the Stokes law-based update of the magnetic field. It also provides edge-centered magnetic field components for the Stokes law-based update of the electric field. The update strategy ensures that the electric field is always consistent with Gauss' law and the magnetic field is always divergence-free. This collocation also ensures that electromagnetic radiation that is propagating in a vacuum has both electric and magnetic fields that are exactly divergence-free. Coupled relativistic fluid dynamic equations are solved for the positively and negatively charged fluids. The fluids' numerical fluxes also provide a self-consistent current density for the update of the electric field. Our reconstruction strategy ensures that fluid velocities always remain sub-luminal. Our third innovation consists of an efficient design for several popular IMEX schemes so that they provide strong coupling between the finite-volume-based fluid solver and the electromagnetic fields at high order. This innovation makes it possible to efficiently utilize high order IMEX time update methods for stiff source terms in the update of high order finite-volume methods for hyperbolic conservation laws. We also show that this very general innovation should extend seamlessly to Runge–Kutta discontinuous Galerkin methods. The IMEX schemes enable us to use large CFL numbers even in the presence of stiff source terms. Several accuracy analyses are presented showing that our method meets its design accuracy in the MHD limit as well as in the limit of electromagnetic wave propagation. Several stringent test problems are also presented. We also present a relativistic version of the GEM problem, which shows that our algorithm can successfully adapt to challenging problems in high energy astrophysics. [ABSTRACT FROM AUTHOR]

For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, namely the weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and discontinuous Galerkin (DG) finite element methods, emphasizing several of their recent developments: bound-preserving limiters for DG, finite volume and finite difference schemes, which address issues in robustness and accuracy; WENO limiters for DG methods, which address issues in non-oscillatory performance when there are strong shocks, and inverse Lax–Wendroff type boundary treatments for finite difference schemes, which address issues in solving complex geometry problems using Cartesian meshes. [ABSTRACT FROM AUTHOR]

We present a methodology for the high order approximation of hyperbolic conservation laws in networks by using the Dumbser–Enaux–Toro solver and exact solvers for the classical Riemann problem at junctions. The proposed strategy can be applied to any hyperbolic system, conservative or non-conservative, and possibly with flux functions containing discontinuous parameters, as long as an exact or approximate Riemann problem solver is available. The methodology is implemented for a one-dimensional blood flow model that considers discontinuous variations of mechanical and geometrical properties of vessels. The achievement of formal order of accuracy, as well as the robustness of the resulting numerical scheme, is verified through the simulation of both, academic tests and physiological flows. [ABSTRACT FROM AUTHOR]

A high-order Residual Distribution scheme for the solution of the compressible RANS equations is developed. The one-equation Spalart–Allmaras turbulence model is solved together with the RANS equations as a fully coupled system, in which both the mean flow equations and the turbulence equation are solved with high-order of accuracy. A continuous approximation of the solution is adopted and standard Lagrangian basis functions are used to construct the discrete space. Since viscous terms involve the gradient of the numerical solution which has a discontinuous normal component across the faces of the elements, a continuous approximation of the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same basis functions used for the solution. The non-linear system of equations resulting from the numerical discretization is solved with the non-linear LU-SGS method. Several numerical experiments are performed to verify the accuracy of the numerical method, these are based on both subsonic and transonic flows in two and three spatial dimensions. [ABSTRACT FROM AUTHOR]

A robust and high order accurate Residual Distribution (RD) scheme for the discretization of the steady Navier–Stokes equations is presented. The proposed method is very flexible: it is formulated for unstructured grids, regardless the shape of the elements and the number of spatial dimensions. A continuous approximation of the solution is adopted and standard Lagrangian shape functions are used to construct the discrete space, as in Finite Element methods. The traditional technique for designing RD schemes is adopted: evaluate, for any element, a total residual, split it into nodal residuals sent to the degrees of freedom of the element, solve the non-linear system that has been assembled and then iterate up to convergence. The main issue addressed by the paper is that the technique relies in depth on the continuity of the normal flux across the element boundaries: this is no longer true since the gradient of the state solution appears in the flux, hence continuity is lost when using standard finite element approximations. Naive solution methods lead to very poor accuracy. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, a continuous approximation of the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution, preserving the optimal accuracy of the method. Linear and non-linear schemes are constructed, and their accuracy is tested with the method of the manufactured solutions. The numerical method is also used for the discretization of smooth and shocked laminar flows in two and three spatial dimensions. [ABSTRACT FROM AUTHOR]

• A discretely (locally and globally) conservative high-order semi-discrete entropy-stable scheme has been presented. • The scheme conserves mass, momentum, energy, and satisfies the celebrated H-theorem at a discrete level. • To address low/high rarefaction flows, we couple the scheme with an iteration-free implicit-explicit time discretization. • The scheme yields stable solution for Riemann problems involving shocks even without using a shock-limiter. The kinetic equations govern the behavior of gaseous flows, compressibility, turbulence, reactions with internal energy exchange, that form the essence of numerous physical processes. Despite their wide applicability, their six-dimensional (seven including time) nature presents a huge computational challenge. With the advent of the modern high performance computing systems and few recent advances in numerical methods, it is now possible to numerically study the behavior of these systems. However, to understand the instability of rich molecular processes and find a structure in the apparent chaos, a scheme that is efficient in multi-dimensions, exhibits high parallel-efficiency, and foremost produces an entropy solution as per the theory of solutions of hyperbolic systems may prove useful. In this work, first, we construct an entropy stable flux for non-linear inhomogeneous (full) Boltzmann equation. Second, to ensure geometrical flexibility, we couple the scheme with a class of high order discontinuous Galerkin discretization (Jaiswal 2019 [41]) which satisfy summation-by-part (SBP) discretely. Third, utilizing SBP, we prove that the resulting semi-discrete scheme is locally and globally conservative in flat spaces; preserves the entropy-decay (H-theorem) property; efficient and simple; and therefore suitable for dealing with highly complex non-smooth flow problems. Fourth, we show that the fully-discrete kinetic scheme, utilizing an implicit-explicit time-discretization, in the limit of vanishing Knudsen number, becomes an entropy-stable explicit scheme applied to the Euler system. Fifth, we carry out a series of verification tests to illustrate the stability, accuracy, and conservation properties of the proposed method. These tests involve over 85 million degrees of freedom and over 50 billion operations per time-step. [ABSTRACT FROM AUTHOR]

Abstract: The goal of this paper is to formulate genuinely multidimensional HLL and HLLC Riemann solvers for unstructured meshes by extending our prior papers on the same topic for logically rectangular meshes Balsara (2010, 2012) [4,5]. Such Riemann solvers operate at each vertex of a mesh and accept as an input the set of states that come together at that vertex. The mesh geometry around that vertex is also one of the inputs of the Riemann solver. The outputs are the resolved state and multidimensionally upwinded fluxes in both directions. A formulation which respects the detailed geometry of the unstructured mesh is presented. Closed-form expressions are provided for all the integrals, making it particularly easy to implement the present multidimensional Riemann solvers in existing numerical codes. While it is visually demonstrated for three states coming together at a vertex, our formulation is general enough to treat multiple states (or zones with arbitrary geometry) coming together at a vertex. The present formulation is very useful for two-dimensional and three-dimensional unstructured mesh calculations of conservation laws. It has been demonstrated to work with second to fourth order finite volume schemes on two-dimensional unstructured meshes. On general triangular grids an arbitrary number of states might come together at a vertex of the primal mesh, while for calculations on the dual mesh usually three states come together at a grid vertex. We apply the multidimensional Riemann solvers to hydrodynamics and magnetohydrodynamics (MHD) on unstructured meshes. The Riemann solver is shown to operate well for traditional second order accurate total variation diminishing (TVD) schemes as well as for weighted essentially non-oscillatory (WENO) schemes with ADER (Arbitrary DERivatives in space and time) time-stepping. Several stringent applications for compressible gasdynamics and magnetohydrodynamics are presented, showing that the method performs very well and reaches high order of accuracy in both space and time. The present multidimensional Riemann solver is cost-competitive with traditional, one-dimensional Riemann solvers. It offers the twin advantages of isotropic propagation of flow features and a larger CFL number. Please see http://www.nd.edu/~dbalsara/Numerical-PDE-Course for a video introduction to multidimensional Riemann solvers. [Copyright &y& Elsevier]

This paper studies higher-order finite volume methods for solving elliptic boundary value problems. We develop a general framework for construction and analysis of higher-order finite volume methods. Specifically, we establish the boundedness and uniform ellipticity of the bilinear forms for the methods, and show that they lead to an optimal error estimate of the methods. We prove that the uniform local-ellipticity of the family of the bilinear forms ensures its uniform ellipticity. We then establish necessary and sufficient conditions for the uniform local-ellipticity in terms of geometric requirements on the meshes of the domain of the differential equation, and provide a general way to investigate the mesh geometric requirements for arbitrary higher-order schemes. Several useful examples of higher-order finite volume methods are presented to illustrate the mesh geometric requirements. [ABSTRACT FROM AUTHOR]

Abstract: The finite volume discretisation of the shallow water equations has been the subject of many previous studies, most of which deal with a well-balanced conservative discretisation of the convective flux and bathymetry. However, the bed friction discretisation has not been so profusely analysed in previous works, while it may play a leading role in certain applications of shallow water models. In this paper we analyse the numerical discretisation of the bed friction term in the two-dimensional shallow water equations, and we propose a new unstructured upwind finite volume discretisation for this term. The new discretisation proposed improves the accuracy of the model in problems in which the bed friction is a relevant force in the momentum equation, and it guarantees a perfect balance between gravity and bed friction under uniform flow conditions. The relation between the numerical scheme used to solve the hydrodynamic equations and the scheme used to solve a scalar transport model linked to the shallow water equations, is also analysed in the paper. It is shown that the scheme used in the scalar transport model must take into consideration the scheme used to solve the hydrodynamic equations. The most important implication is that a well-balanced and conservative scheme for the scalar transport equation cannot be formulated just from the water depth and velocity fields, but has to consider also the way in which the hydrodynamic equations have been solved. [Copyright &y& Elsevier]

Abstract: We construct a third order multidimensional upwind residual distribution scheme for the system of the Navier–Stokes equations. The underlying approximation is obtained using standard P 2 Lagrange finite elements. To discretise the inviscid component of the equations, each element is divided in sub-elements over which we compute a high order residual defined as the integral of the inviscid fluxes on the boundary of the sub-element. The residuals are distributed to the nodes of each sub-element in a multi-dimensional upwind way. To obtain a discretisation of the viscous terms consistent with this multi-dimensional upwind approach, we make use of a Petrov–Galerkin analogy. The analogy allows to find a family of test functions which can be used to obtain a weak approximation of the viscous terms. The performance of this high-order method is tested on flows with high and low Reynolds number. [Copyright &y& Elsevier]

Abstract: In this paper we consider the very high order approximation of solutions of the Euler equations. We present a systematic generalization of the residual distribution method of to very high order of accuracy, by extending the preliminary work discussed in to systems and hybrid meshes. We present extensive numerical validation for the third and fourth order cases with Lagrange finite elements. In particular, we demonstrate that we both have a non-oscillatory behavior, even for very strong shocks and complex flow patterns, and the expected accuracy on smooth problems. [Copyright &y& Elsevier]

We develop a very simple algorithm that permits to construct compact, high order schemes for steady first order Hamilton-Jacobi equations. The algorithm relies on the blending of a first order scheme and a compact high order scheme. The blending is conducted in such a way that the scheme is formally high order accurate. A convergence proof without error estimate is given. We provide several numerical illustrations that demonstrate the effective accuracy of the scheme. The numerical examples use triangular unstructured meshes, but our method may be applied to other kind of meshes. Several implementation remarks are also given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the numerical approximation of bedload sediment transport due to shallow layer flows. The hydrodynamical component is modeled by a 2D shallow water system and the morphodynamical component by a solid transport discharge formula that depends on the hydrodynamical variables. The coupled system can be written as a nonconservative hyperbolic system. To discretize it, first we consider a Roe-type first order scheme as well as a variant based on the use of flux limiters. These first order schemes are then extended to second order accuracy by means of a new MUSCL-type reconstruction operator on unstructured meshes. Finally, some numerical tests are presented. [Copyright &y& Elsevier]

Abstract: To simulate turbulent flows in complicated enclosed three-dimensional domains a fast finite-volume high-order method is developed. In principle, the method is based on the Chorin–Temam scheme. The Poisson solver, which is applied to compute the pressure, uses the separation of variables together with capacitance matrix technique. The developed numerical method generally allows to use hexahedral computational meshes, which are non-equidistant in all three directions and non-regular in any two directions. The method was successfully used in three-dimensional Direct Numerical Simulations of turbulent high-Rayleigh-number thermal convection in cylindrical and parallelepiped domains with obstacles. [Copyright &y& Elsevier]

This study investigates the instability issue at a free surface when the consistent schemes based on variable differences are applied in semi-implicit particle methods. A semi-analytical error-analysis method is proposed to clarify how the incomplete/biased neighbor support triggers error accumulation and instability. Specifically, the discretization models are decomposed into the center-variable components (CVCs) and neighbor-variable components (NVCs). The influence of different components on error accumulation is analyzed theoretically and numerically. Based on the error analysis, new indices are proposed to evaluate the risk of error accumulation due to the biased neighbor support. Then, novel free-surface-detection conditions are proposed from the indices by detecting the particles prone to error accumulation as free surface particles. Numerical examples demonstrated that the proposed conditions detected fewer free-surface particles but produced more stable simulations compared to the existing conditions. • Numerical instability of consistent schemes at free surfaces is investigated. • The mechanism of error accumulation due to biased particle distributions is discovered. • Novel indices are proposed to evaluate error-accumulation risk and detect free surfaces. • Numerical examples demonstrated the effectiveness of proposed indices for stability. [ABSTRACT FROM AUTHOR]

We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the solution, and we establish their energy stability. This class includes as a special case high order, unconditionally stable schemes obtained via convexity splitting. The new schemes are demonstrated on a variety of gradient flows, including partial differential equations that are gradient flow with respect to the Wasserstein (mass transport) distance. • High-order, energy stable, semi-implicit schemes for general gradient flows. • The schemes cost as much as multiple calls per time step to a first-order scheme. • Schemes for gradient flows with varying inner products, such as Wasserstein. • Numerical examples of the new schemes on a variety of gradient flows. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a theoretical analysis and two applications of a high-order overlapping grid method for coupling Cartesian and curvilinear grids, developed in order to simulate aerodynamic noise. First, the overlapping grid method based on Lagrange interpolating polynomials is described and a theoretical analysis of the interpolation operator is then carried out. It shows that the interpolation generates spurious modes that depend on the wavenumbers of the signal. Besides it also gives the optimal conditions in which interpolation can be applied. Then an application to the simulation of the aeroacoustic noise generated by the vortex shedding behind a cylinder is presented. During this simulation, it appears that interpolation can create some spurious acoustic modes in regions where hydrodynamic fluctuations are significant, as predicted by the theoretical analysis. It is shown that these spurious modes disappear when a refined Cartesian grid is used (26 points per wavelength of the vortex shedding were found to be adequate in this study). At last, the simulation of the aerodynamic noise of a three element high-lift wing profile has then been carried out. For this application, the main acoustic source at the slat trailing edge is represented analytically. The propagation of the generated acoustic wave is simulated with a mean flow at rest and with a steady turbulent mean flow computed by RANS. The first application allows us to assess the method by comparing the results to a reference solution. The second one shows that the influence of a non-uniform mean flow on the directivity of an acoustic source can be observed in complex geometries. This application therefore shows that the proposed coupling method is well adapted to complex geometries that are usually met in industrial applications. [Copyright &y& Elsevier]

This paper describes a solution technique for multidimensional elliptic problems based on the use of some third order nodal finite elements and on a reduction of the basic (multidimensional) problem to a set of coupled one-dimensional problems. This solution technique, developed rather heuristically in the framework of nuclear reactor computations in conjunction with early nodal methods, gets on a much firmer ground when applied with nodal finite elements. The first part of the paper deals with the general context of variational nodal finite element methods. The so-called “Transverse and Reduced Integration Method” is then described in the second part of the paper. Its numerical properties are illustrated by some examples. [Copyright &y& Elsevier]

For energetic flows there are many advantages of high order schemes over low order schemes. Here we examine a previously unknown advantage. It is commonly thought that the number of points per wavelength in order to obtain a given error in a numerical approximation depends only on runtime and the order of the approximation. Using truncation error arguments and examples we will show that it is not a constant and depends also on the wavenumber. This dependence on the numerical order and wavenumber strongly favors high order schemes for use in flows which have significant energy in the high modes such at Rayleigh–Taylor and Richtmyer–Meshkov instabilities. [ABSTRACT FROM AUTHOR]

In this paper, we consider several high-order schemes in one space dimension. In particular, we compare the second-order relaxation (∈ << 1) or ‘relaxed’ (∈ = 0) schemes of Jin and Xin [Comm. Pure Appl. Math., 48 (1995), pp. 235–277] with the second-order Lax­Friedrichs scheme of Nessyahu and Tadmor [J. Comp. Phys., 87 (1990), pp. 408–463] and with higher-order essentially nonoscillatory (ENO) and weighted essentially nonoscillatory (WENO) schemes. This comparison is made first on a Sod shock tube and then on a very pathological example of a p-system constructed by Greenberg and Rascle [Arch. Ration. Mech. Anal., 115 (1991), pp. 395–407]. This exotic system admits a family of periodic solutions which are shock-free but present pairs of interacting centered compression waves. Therefore, the exact solution contains big spikes. We show how these different schemes face this numerical challenge. [ABSTRACT FROM AUTHOR]