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• A novel positivity-preserving algorithm for implicit NS solver. • A residual correction to compute the correction factor. • A flux correction to enforce the positivity of the solution conservatively. • Positivity-preserving combined with implicit iterations. • Numerical experiments to verify the positivity-preserving capability. This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes that solve compressible Euler and Navier-Stokes equations to ensure the positivity of density and internal energy (or pressure). Previous positivity-preserving algorithms are mainly based on the slope limiting or flux limiting technique, which rely on the existence of low-order positivity-preserving schemes. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes since it is difficult to know if a low-order implicit scheme is positivity-preserving. In the present paper, a new positivity-preserving algorithm is proposed in terms of the flux correction technique. And the factors of the flux correction are determined by a residual correction procedure. For a finite volume scheme that is capable of achieving a converged solution, we show that the correction factors are in the order of unity with additional high-order terms corresponding to the spatial and temporal rates of convergence. Therefore, the proposed positivity-preserving algorithm is accuracy-reserving and asymptotically consistent. The notable advantage of this method is that it does not rely on the existence of low-order positivity-preserving baseline schemes. Therefore, it can be applied to the implicit schemes solving Euler and especially Navier-Stokes equations. In the present paper, the proposed technique is applied to an implicit dual time-stepping finite volume scheme with temporal second-order and spatial high-order accuracy. The present positivity-preserving algorithm is implemented in an iterative manner to ensure that the dual time-stepping iteration will converge to the positivity-preserving solution. Another similar correction technique is also proposed to ensure that the solution remains positivity-preserving at each sub-iteration. Numerical results demonstrate that the proposed algorithm preserves positive density and internal energy in all test cases and significantly improves the robustness of the numerical schemes. [ABSTRACT FROM AUTHOR]

This paper presents high-order finite volume multi-resolution weighted essentially non-oscillatory schemes with adaptive linear weights to solve hyperbolic conservation laws on triangular meshes. They are abbreviated as the ALW-MR-WENO schemes. The novel third-order, fourth-order, and fifth-order ALW-MR-WENO schemes are designed by applying two unequal-sized hierarchical central stencils in comparison to the classical WENO schemes which utilize many equal-sized upwind biased/central stencils. With the application of one simple condition, only two linear weights are automatically adjusted to be positive values on condition that their summation is one. The novel finite volume ALW-MR-WENO schemes could maintain the designed order of accuracy in smooth areas and reduce to the first-order accuracy so as to keep essentially non-oscillatory properties around strong discontinuities. So it is the first time that any high-order WENO schemes with the application of only two unequal-sized stencils are obtained on triangular meshes. And the major benefits are their efficiency, compactness, and simplicity in large scale engineering applications on unstructured meshes. Finally, several tests are used to indicate the effectiveness of these new finite volume WENO schemes. • We design high-order finite volume multi-resolution weighted essentially non-oscillatory schemes with adaptive linear weights on triangular meshes. They use only two unequal-sized central spatial stencils to obtain arbitrarily high-order spatial approximations in smooth regions. • Two linear weights that sum to one can be automatically adjusted to be any positive values with one simple condition on triangular meshes. • Their major benefits are the high efficiency, compactness, robustness, and simplicity in large scale engineering applications on triangular meshes. [ABSTRACT FROM AUTHOR]

Efficient numerical schemes for solving the Poisson equation with jump conditions are of great interest for a variety of problems, including the modeling of electroporation phenomena and filamentary discharges. In this paper, we propose a modification to a finite volume scheme, namely the discrete dual finite volume method, in order to account for jump conditions with surface charges, i.e. with a source term. Our numerical tests demonstrate second-order convergence even with highly distorted meshes. We then apply the proposed method to model electroporation phenomena in biological cells by proposing a model that considers the thickness of the cell membrane as a separate domain, which differs from the literature. We show the advantages of the proposed method in this context through numerical experiments. • The discrete dual finite volume scheme is extended to solve the Poisson equation with jump conditions and surface charges. • The method is shown to exhibit a second-order convergence through canonical numerical tests. • The method is applied to the electroporation phenomena, where accurate modeling of the potential at the membrane is obtained. • Numerical experiments on the stationary and non-stationary case are provided. [ABSTRACT FROM AUTHOR]

We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L 2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori , each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field. • Semi-implicit FV/FE method for incompressible viscous and resistive MHD equations. • Well-balanced and exactly divergence-free on general unstructured mixed-element grids. • Constrained L2 projection for an exactly divergence-free reconstruction. • Thorough study of the lid-driven MHD cavity problem (reference solution is provided). • Stable long-time simulation of Grad-Shafranov equilibria in 3D tokamak geometries. [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]

We propose high-resolution numerical schemes for two-phase compressible flow simulations to reproduce dynamically created gas/vapor-liquid interfaces with phase change. In the Godunov-type finite volume framework, suppressing numerical dissipation errors in numerical schemes is crucial for capturing the discontinuous solutions. The MUSCL scheme has second-order accuracy for smooth solutions and non-oscillatory behavior near discontinuous solutions. However, the MUSCL scheme introduces excessive numerical dissipation and diffuses the discontinuities nonphysically, leading to difficulties in distinguishing gas/vapor and liquid phases, and thus causing a blur in the interfaces during the multi-phase flow simulations. The hybrid-type boundary variation diminishing (BVD) scheme in this paper combines the MUSCL scheme and the THINC scheme to reduce the numerical dissipation errors near the discontinuities. The MUSCL-THINC-BVD scheme applies the MUSCL scheme for smooth solutions and the THINC scheme for discontinuous solutions, resulting in the successful capture of the discontinuities including the dynamically created gas/vapor-liquid interfaces. The Adaptive THINC-BVD scheme, which switches two types of THINC schemes with different values of gradient parameter, also captures the discontinuities clearly. The numerical results of the benchmark tests show that the proposed BVD schemes can lucidly reproduce the vapor-liquid interfaces newly created during the dynamical process of phase change. • Hybrid BVD schemes are presented to compute gas/vapor interfaces in 6-equation model for multi-phase flows. • Interfaces newly generated between liquid and vapor during the evaporation process are well-reproduced for the first time. • Basic benchmark validations are conducted for two-phase flows associated with phase changes. • The numerical methods presented are well-suited for simulating multi-phase flows involving phase changes. [ABSTRACT FROM AUTHOR]

In this paper, the new distinct-positivity-preserving (DPP) methods are proposed for fifth-order finite volume multi-resolution WENO schemes with adaptive linear weights (which are termed as the ALW-MR-WENO schemes) [J. Comput. Phys., 493 (2023), 112471] for solving some extreme problems on structured meshes. Associated WENO spatial reconstruction procedures only require two hierarchical unequal-sized central spatial stencils for achieving fifth-order accuracy in smooth regions and keeping the essentially non-oscillatory property in non-smooth regions. One redefines five cell averages after performing such spatial reconstructions and then designs one quartic polynomial and two cubic reconstruction polynomials based on them in one dimension. After that, a new detective process is used to examine the positivity of the point values of such cubic reconstruction polynomials at some checking points inside the target cell. If the negativity happens, a new over-determined compression limiter is carried out to ensure that the compressed polynomials can achieve the positivity in the whole target cell instead of only at some discrete points inside it. The quartic polynomial could use its four quadrature point values by using a four-point Gauss-Lobatto quadrature formula when performing numerical integrations. Since it is easy to rewrite the point values of one quartic polynomial with the point values of two cubic polynomials timing with associated linear weights, both cubic polynomials could use a four-point Gauss-Lobatto quadrature formula or a three-point Simpson's quadrature formula without losing any algebra precisions. By doing this novelty switching methodology between two quadrature formulas, it is the first time to theoretically prove and increase a sufficient CFL number from 1/12 to 1/7.2 for the fifth-order WENO schemes in one dimension. This methodology can be easily extended to multi-dimensions. Finally, the novelty DPP methods for fifth-order ALW-MR-WENO schemes with a larger practical CFL number of 0.6, instead of the sufficient CFL number of 1/7.2, are also simple and robust enough for some extreme problems without timely halving the CFL number to avoid the appearance of negative density and negative pressure. [ABSTRACT FROM AUTHOR]

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]

In this paper we propose a fluctuation splitting finite volume scheme for a non-conservative modeling of shear shallow water flow (SSWF). This model was originally proposed by Teshukov (2007) in [14] and was extended to include modeling of friction by Gavrilyuk et al. (2018) in [7]. The directional splitting scheme proposed by Gavrilyuk et al. (2018) in [7] is tricky to apply on unstructured grids. Our scheme is based on the physical splitting in which we separate the characteristic waves of the model to form two different hyperbolic sub-systems. The fluctuations associated with each sub-systems are computed by developing Riemann solvers for these sub-systems in a local coordinate system. These fluctuations enable us to develop a Godunov-type scheme that can be easily applied on mixed/unstructured grids. While the equation of energy conservation is solved along with the SSWF model in [7] , in this paper we solve only SSWF model equations. We develop a cell-centered finite volume code to validate the proposed scheme with the help of some numerical tests. As expected, the scheme shows first order convergence. The numerical simulation of 1D roll waves shows a good agreement with the experimental results. The numerical simulations of 2D roll waves show similar transverse wave structures as observed in [7]. • A finite volume scheme is proposed for non-conservative shear shallow water equations. • A preferable 'family of paths' is used to design approximate Riemann solvers. • These Riemann solvers are used in to design a fluctuation based Godunov-type scheme. • The proposed scheme is validated by performing some numerical tests. • The scheme can be a useful tool to study the shear shallow water equations. [ABSTRACT FROM AUTHOR]

Nowadays numerical methods for handling partial differential equations is so vast that it is almost taken for granted. In this paper, we propose a discrete duality finite volume scheme with biquadratic elements and its application in solving two-dimensional elliptic equation on quadrilateral meshes. To test the robustness and efficiency of this method, we investigate several examples from physical and engineering fields with different parameters and coefficients. Numerical experiments, encompassing linear and nonlinear elliptic equations with constant or variable coefficients, reveal that the new method achieves optimal convergence in the continuous H 1 -norm, with a 2nd-order convergence, and in the L 2 -norm, with a 3rd-order convergence. Moreover, comparison with the existing biquadratic element finite volume scheme shows that the above method has an advantage of accuracy in the discrete L ∞ -norm on several distorted meshes. [ABSTRACT FROM AUTHOR]

The recently-developed general synthetic iterative scheme (GSIS) is efficient in simulating multiscale rarefied gas flows due to the coupling of mesoscopic kinetic equation and macroscopic synthetic equation: for linearized Poiseuille flow where the boundary flux is fixed at each iterative step, the steady-state solutions are found within dozens of iterations in solving the gas kinetic equations, while for general nonlinear flows the iteration number is increased by about one order of magnitude, caused by the incompatible treatment of the boundary flux for the macroscopic synthetic equation. In this paper, we propose a generalized boundary treatment (GBT) to further accelerate the convergence of GSIS. The main idea is, the truncated velocity distribution function at the boundary, similar to that used in the Grad 13-moment equation, is reconstructed by the macroscopic conserved quantities from the synthetic equation, as well as the high-order correction of non-equilibrium stress and heat flux from the kinetic equation; therefore, in each inner iteration solving the synthetic equation, the explicit constitutive relations facilitate real-time updates of the macroscopic boundary flux, driving faster information exchange in the flow field, and consequently achieving faster convergence. Moreover, the high-order correction derived from the kinetic equation can compensate the approximation by the truncation and ensure the boundary accuracy. The one-dimensional Fourier flow, two-dimensional hypersonic flow around a cylinder, three-dimensional pressure-driven pipe flow and the flow around the hypersonic technology vehicle are simulated. The accuracy of GSIS-GBT is validated by the direct simulation Monte Carlo method, the previous versions of GSIS, and the unified gas-kinetic wave-particle method. For the efficiency, in the near-continuum flow regime and slip regime, GSIS-GBT can be faster than the conventional iteration scheme in the discrete velocity method and the previous versions of GSIS by two- and one-order of magnitude, respectively. • The generalized boundary treatment (GBT) is proposed to further accelerate the general synthetic iterative scheme (GSIS). • In both low-speed internal flows and hypersonic flows, GSIS-GBT can achieve convergence within 100 iterations. • GSIS-GBT is respectively faster than previous version of GSIS/conventional iterative scheme by one/two orders of magnitude. [ABSTRACT FROM AUTHOR]

Accurate and efficient numerical simulations of soil water movement, as described by the highly nonlinear Richards' equation, often require local refinement near recharge or sink/source terms. In this paper, we present a novel numerical scheme for solving Richards' equation. Our approach is based on the vertex-centered finite volume method (VCFVM) and can be easily adapted to locally refined meshes. The proposed scheme offers some key features, including the definition of all unknowns over vertices of the primary mesh, expression of flux crossing dual edges as combinations of hydraulic heads at the vertices of the primary cell, and the capability to handle nonmatching meshes in the presence of local mesh refinement. For performance evaluation, soil water content and soil water potential simulated by the proposed scheme are benchmarked against results produced from HYDRUS (a widely used soil water numerical model) and the observed values in four test cases, including a convergence test case, a synthetic case, a laboratory experiment case and a field experiment case. The comparison results demonstrate the effectiveness and applicability of our scheme across a wide range of soil parameters and boundary conditions. [ABSTRACT FROM AUTHOR]

Fourier analysis based optimization techniques have been developed for numerical schemes on structured grids and result in significant performance improvements. However, there lacks an effective optimization technique for numerical schemes on general unstructured grids, due to the mesh geometry complexity that makes Fourier transformation infeasible. To address this issue, an optimization framework based on machine learning is developed in this paper for numerical schemes on unstructured grids. The optimization of a compact high-order variational reconstruction on triangular grids is performed to illustrate the framework. An artificial neural network (ANN) is used to predict the optimal values of the derivative weights on cell interfaces that are the free parameters of the variational reconstruction, given the local geometry as input. The ANN is trained by minimizing the solution errors of a linear advection equation on a set of generated compact stencils, with a set of sampled sine waves as initial conditions. The developed optimization framework is applicable to general numerical schemes on unstructured grids as the training does not require explicit computation of dispersion or dissipation. Numerical results for inviscid flow problems show that the optimized variational finite volume scheme is remarkably more accurate and efficient than its non-optimized counterpart, demonstrating the effectiveness of the machine learning optimization. • A machine learning optimization framework for numerical schemes on unstructured grids. • A feedforward neural network to predict optimal parameter values of numerical schemes. • Training of the network by minimizing solution errors of a linear advection equation. • A machine learning optimized compact high-order variational reconstruction. • Generalization of the optimized variational reconstruction to nonlinear problems. [ABSTRACT FROM AUTHOR]

This paper introduces a new approach to constructing algorithms for gradient computations on unstructured grids. The proposed approach is to derive a gradient algorithm from a hyperbolic diffusion scheme, which solves the diffusion equation and equations for the solution gradient, by ignoring the residual component for the diffusion equation. The resulting scheme forms a globally coupled linear system of equations for the gradients with the Green–Gauss gradient formula on the right hand side, which can be solved efficiently by iterative techniques. Accuracy and iterative performance can be controlled by parameters inherited from the generating hyperbolic diffusion scheme. Second- or fourth-order gradient accuracy can be obtained on regular grids, and first-order accuracy, which is sufficient for second-order finite-volume schemes, can be obtained on irregular grids. Numerical results indicate that the method gives highly accurate gradients especially for highly-curved high-aspect-ratio grids typical in high-Reynolds-number boundary layer flows over complex geometries, and can be employed in finite-volume solvers as an efficient alternative to least-squares or Green–Gauss gradient methods. It is also shown that one of the parameters has a role of dissipation, and can be tuned to yield a smooth gradient distribution over a discontinuous function, thereby allowing an implicit finite-volume solver to converge for a discontinuous solution with a mild overshoot. The paper concludes with a discussion for future developments. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modification of an acoustic-transport operator splitting Lagrange-projection method for simulating compressible flows with gravity. The original method involves two steps that respectively account for acoustic and transport effects. Our work proposes a simple modification of the transport step, and the resulting modified scheme turns out to be a flux-splitting method. This new numerical method is less computationally expensive in the low-Mach regime, more memory efficient, and easier to implement than the original one. We prove stability properties for this new scheme by showing that under classical CFL conditions, the method is positivity preserving for mass, energy and entropy satisfying. The flexible flux-splitting structure of the method enables straightforward extensions of the method to multi-dimensional problems (with respect to space) and high-order discretizations that are presented in this work. We also propose an interpretation of the flux-splitting solver as a relaxation approximation. Both the stability and the accuracy of the new method are tested against one-dimensional and two-dimensional numerical experiments that involve highly compressible flows and low-Mach regimes. • The all-regime acoustic/transport splitting solver proposed in [16] can be modified into a flux-splitting, one-step version. • The resulting method is less computationally expensive, more memory efficient, and easier to implement than the original one. • Stability properties are derived by interpreting the scheme as a convex combination of the advection/pressure steps. • This recasting allows an easy high-order extension as the new method is a standard flux-based solver. • Following the lines of [32] , we equip our scheme with the well-balanced property for gravity source terms. [ABSTRACT FROM AUTHOR]

This paper studies three high-order structure-preserving finite volume weighted essentially non-oscillatory (WENO) methods, which are not only well balanced (WB) for a general known hydrostatic equilibrium state but also preserve the positivity of density and pressure, for the compressible Euler equations under gravitational fields. These methods are built on a simple local scaling positivity-preserving (PP) limiter and a modified WENO-ZQ reconstruction exactly preserving the cell average value and scaling invariance. The WB properties of these three methods are achieved based on suitable numerical fluxes and approximation to the gravitational source terms. Based on some convex decomposition techniques as well as several critical properties of the admissible states and numerical flux, we carry out rigorous positivity-preserving analyses for these three WB schemes. We rigorously prove that the three WB methods, coupled with the PP limiter and a strong-stability-preserving time discretization, are always PP under suitable Courant-Friedrichs-Lewy conditions. Extensive numerical examples are provided to confirm WB and PP properties of three methods. [ABSTRACT FROM AUTHOR]

The Finite Volume method with Exact two-material Riemann Problems (FIVER) is both a computational framework for multi-material flows characterized by large density jumps, and an Embedded Boundary Method (EBM) for computational fluid dynamics and highly nonlinear Fluid–Structure Interaction (FSI) problems. This paper deals with the EBM aspect of FIVER. For FSI problems, this EBM has already demonstrated the ability to address viscous effects along wall boundaries, and large deformations and topological changes of such boundaries. However, like for most EBMs – also known as immersed boundary methods – the performance of FIVER in the vicinity of a wall boundary can be sensitive with respect to the position and orientation of this boundary relative to the embedding mesh. This is mainly due to ill-conditioning issues that arise when an embedded interface becomes too close to a node of the embedding mesh, which may lead to spurious oscillations in the computed solution gradients at the wall boundary. This paper resolves these issues by introducing an alternative definition of the active/inactive status of a mesh node that leads to the removal of all sources of potential ill-conditioning from all spatial approximations performed by FIVER in the vicinity of a fluid–structure interface. It also makes two additional contributions. The first one is a new procedure for constructing the fluid–structure half Riemann problem underlying the semi-discretization by FIVER of the convective fluxes. This procedure eliminates one extrapolation from the conventional treatment of the wall boundary conditions and replaces it by an interpolation, which improves robustness. The second contribution is a post-processing algorithm for computing quantities of interest at the wall that achieves smoothness in the computed solution and its gradients. Lessons learned from these enhancements and contributions that are triggered by the new definition of the status of a mesh node are then generalized and exploited to eliminate from the original version of the FIVER method its sensitivities with respect to both of the position and orientation of the wall boundary relative to the embedding mesh, while maintaining the original definition of the status of a mesh node. This leads to a family of second-generation FIVER methods whose performance is illustrated in this paper for several flow and FSI problems. These include a challenging flow problem over a bird wing characterized by a feather-induced surface roughness, and a complex flexible flapping wing problem for which experimental data is available. [ABSTRACT FROM AUTHOR]

In Hu and Yi (2016) [20] , a non-oscillatory k -exact reconstruction method was proposed towards the high-order finite volume methods for steady Euler equations, which successfully demonstrated the high-order behavior in the simulations. However, the degeneracy of the numerical accuracy of the approximate solutions to problems with curved boundary can be observed obviously. In this paper, the issue is resolved by introducing the Non-Uniform Rational B-splines (NURBS) method, i.e., with given discrete description of the computational domain, an approximate NURBS curve is reconstructed to provide quality quadrature information along the curved boundary. The advantages of using NURBS include i). both the numerical accuracy of the approximate solutions and convergence rate of the numerical methods are improved simultaneously, and ii). the NURBS curve generation is independent of other modules of the numerical framework, which makes its application very flexible. It is also shown in the paper that by introducing more elements along the normal direction for the reconstruction patch of the boundary element, significant improvement in the convergence to steady state can be achieved. The numerical examples confirm the above features very well. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an approximation method that establishes certain order enhancements by leveraging radial basis functions (RBFs) in the numerical solution of conservation laws. The use of RBFs for interpolation and approximation is a well developed area of research. Of particular interest in this work is the development of high order finite volume (FV) weighted essentially non-oscillatory (WENO) methods, which utilize RBF approximations to obtain required data at cell interfaces. The aforementioned improvement in the order of accuracy is addressed through an analysis of the truncation error, resulting in expressions for the shape parameters appearing in the basis. This paper seeks to address the practical elements of the approach, including the evaluations of shape parameters as well as a hybrid implementation. To highlight the effectiveness of the non-polynomial basis in shock-capturing, the proposed methods are applied to systems of one-dimensional hyperbolic and weakly hyperbolic conservation laws and compared with several well-known WENO schemes in the literature. We also include a two-dimensional example for a scalar problem that demonstrates an extension to multiple dimensions. In the case of the non-smooth, weakly hyperbolic test problem, notable improvements are observed in predicting the location and height of the finite time blowup. The numerical results demonstrate that the proposed schemes attain notable improvements in accuracy, as indicated by the analysis of the reconstructions. A key contribution of this work is the development of robust third-order WENO method, which further demonstrates the effectiveness of the non-polynomial basis. • Order enhanced FV WENO methods which utilize radial basis function(RBF) approximation. • Order of accuracy is addressed through an analysis for the shape parameters in RBFs. • The implementation employs a hybrid strategy, aims to alleviate computational costs. • Applied to hyperbolic and weakly hyperbolic conservation laws with well-known schemes. [ABSTRACT FROM AUTHOR]

• Our Hamiltonian splitting method preserves Poisson structure of Vlasov–Maxwell system. • The Gauss's Law is satisfied by the solution after time discretization. • The evolution of distribution function can be expressed as several one-dimensional translations. In this paper, the numerical discretizations based on Hamiltonian splitting for solving the Vlasov–Maxwell system are constructed. We reformulate the Vlasov–Maxwell system in Morrison–Marsden–Weinstein Poisson bracket form. Then the Hamiltonian of this system is split into five parts, with which five corresponding Hamiltonian subsystems are obtained. The splitting method in time is derived by composing the solutions to these five subsystems. Combining the splitting method in time with the Fourier spectral method and finite volume method in space gives the full numerical discretizations which possess good conservation for the conserved quantities including energy, momentum, charge, etc. In numerical experiments, we simulate the Landau damping, Weibel instability and Bernstein wave to verify the numerical algorithms. [ABSTRACT FROM AUTHOR]

• A parallel and constraint-preserving simulator is proposed for variational inequality solution of the black oil model. • Unlike most existing simulators in literature, our framework keeps the predicted solution within the physical range. • Numerical results show that the proposed active-set reduced-space method is superior to the classical inexact Newton method. • The general methodology can be applicable for the other nonlinear problems in petroleum reservoirs. Due to the rapid advancement of supercomputing resource, there is a growing interest in developing parallel algorithms for the large-scale reservoir simulation. In this paper, we present a parallel and fully implicit simulator for the black oil model based on the variational inequality (VI) framework, which can be used to enforce important mathematical and physical properties to obtain accurate constraint-preserving solutions. In other words, this framework ensures the predicted solution to stay within the physical range. In the proposed approach, the black oil model is reformulated as a variational inequality system that naturally satisfies the basic boundedness requirement of the solution, and then a fully implicit finite volume method is applied to discretize the model equations. In addition to that, a number of nonlinear and linear fast solver technologies, including a variant of inexact Newton methods and the domain decomposition based preconditioners, are employed to guarantee the robustness and parallel scalability of the simulator. A particular emphasis of the proposed framework is placed on the parallel and algorithmic performance of the variational inequality approach across large-scale and heterogeneous problems. Several numerical results pertaining to the problems in one, two and three dimensions are presented to illustrate the efficiency, robustness, and the overall performance of the fully implicit constraint-preserving simulator. [ABSTRACT FROM AUTHOR]

This paper presents a new strategy to couple non-matching interfaces in the Finite Volume Method based on a conservative interpolation. In contrast to most of the conservative methods, the current approach does not modify the mesh and therefore, the connection between variables at both sides of the interfaces is solved using interpolation. The conservation of fluxes is imposed by a pair of vector equations which relates the matrix coefficients of the Finite Volume problem with the weighting factors used for interpolating. These relations are satisfied by projecting the matrix coefficients from an interface to its opposite and then computing the weighting factors according to a conservation principle. This new strategy is implemented for an incompressible flow solver including a transport equation of a scalar function. The conservation and accuracy properties of the proposed methodology are analyzed in a set of numerical problems where the theory and the computational implementation are validated. • A conservative coupling between non-matching interfaces is proposed. • The surface meshes of the non-matching interfaces are left unmodified. • Coefficients of the system of equations are related with the interpolation factors. • The matrix coefficients from one interface are projected into the other one. • The method is validated with several numerical tests showing local conservation. [ABSTRACT FROM AUTHOR]

The article introduces a finite-volume method for the Darcy problem in heterogeneous anisotropic media. The method is based on the mixed formulation for the pressure and its gradient. The method is stable despite collocation of both pressure and its gradient at cell centers and demonstrates the first order convergence on numerous benchmarks as well as good monotonicity property. The method produces quasi-definite matrix, which is numerically shown to have good asymptotics of the condition number. Our flux discretization method is a realization of our more general concept of stable flux discretization for saddle-point systems with vector of several unknowns. In this paper this vector is composed of pressure and its gradient and the saddle-point system is the mixed formulation of the Darcy problem. • A heterogeneous Darcy problem in saddle-point formulation is considered. • A linear finite-volume method with two-point flux approximation is proposed for saddle-point system. • The method is consistent and first order accurate on general meshes with heterogeneous full tensor permeability. • The method is found to satisfy the discrete maximum principle. [ABSTRACT FROM AUTHOR]

• The discrete LS method is developed for two-dimensional time-fractional PDEs on irregular domains. • No mesh division is needed with easy implementation and high accuracy. • The LS method is easy to be extended to three-dimensional problems. • Regularization is applied to reduce the condition number of the coefficient matrix. In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain. [ABSTRACT FROM AUTHOR]

In Hu et al. (2016) [22] , it has been successfully demonstrated that using lower-order moment model correction is a promising idea to accelerate the steady-state computation of high-order moment models of the Boltzmann equation. To develop the existing solver, the following aspects are studied in this paper. First, the finite volume method with linear reconstruction is employed for high-resolution spatial discretization of the resulting equations. Second, by introducing an appropriate parameter τ in the correction step, it is found that the performance of the solver can be improved significantly, i.e., more levels would be involved in the solver, which further accelerates the convergence of the method. Third, Heun's method is employed as the smoother in each level to enhance the robustness of the solver. Numerical experiments in microflows are carried out to demonstrate the efficiency and to investigate the behavior of the new solver. In addition, several order reduction strategies for the choice of the order sequence of the solver are tested, and the strategy m l − 1 = ⌈ m l / 2 ⌉ is found to be most efficient. • A nonlinear multi-level moment solver is proposed for high-order moment models. • The efficiency of the solver is improved significantly without loss of accuracy. • The behavior of the solver is investigated in detail by three microflow benchmarks. • The order reduction strategy m l − 1 = ⌈ m l / 2 ⌉ is found to be most efficient in our tests. [ABSTRACT FROM AUTHOR]

• A diffusion scheme based on a nonlinear combination of two-side flux, instead of one-side flux, is constructed. • It is an improvement of the TPFA scheme by adding a nonlinear tangential component in the construction of discrete flux. • It satisfies the maximum principle and has a minimal stencil on polygonal meshes. • Numerical results show that it is second order accurate on 2D skewed meshes for anisotropic diffusion problems. A new diffusion scheme satisfying the maximum principle on two-dimensional (2D) polygonal meshes is proposed. Being different from the existing schemes which are based on the nonlinear combination of the two one-side fluxes, the scheme in this paper is constructed by a modification of the conventional linear Two-Point Flux Approximation (TPFA) scheme, in which the two-side flux is constructed. To improve its accuracy on 2D skewed meshes, the TPFA scheme is rewritten by adding a tangential part, i.e., the tangential component of the flux multiplied with a nonlinear factor. A new method for the discretization of the tangential part is presented so that both the conservation condition and the maximum principle are satisfied, while the TPFA part remains unchanged. Moreover, by virtue of the harmonic averaging points for the calculation of the tangential part, the scheme has a minimal stencil, similar to the TPFA scheme. Finally, some numerical examples are presented to verify that it is second order accurate on 2D polygonal meshes for anisotropic diffusion problems. [ABSTRACT FROM AUTHOR]

In this paper, a novel Roe-type Riemann solver is developed for solving incompressible two-phase flow problems. The two-phase flow is modeled by coupling the artificial compressibility based Navier-Stokes equations and an interface advection equation in a conservative level set framework. The proposed Roe solver does not assume a locally constant density field while evaluating the flux Jacobian. As a result, the solver is capable of capturing the exact jump conditions across the contact and shear waves, which in turn helps in enhancing the accuracy of the interface motion. Moreover, mass conservation characteristics are significantly improved by adopting conservative level set framework together with finite volume discretization of the system of equations. A dual time stepping method with explicit pseudo-time marching scheme is used for modeling the unsteady terms. The efficacy of the proposed solver is demonstrated using various standard incompressible two-phase flow problems, involving low amplitude sloshing, broken dam, Rayleigh-Taylor instability and droplet splash. • Developed a new Roe-type Riemann solver for incompressible two-phase flow problems. • Numerical results are in good agreement with analytical and experimental results. • Comparison with HLL and other existing Roe solvers is presented. • The proposed Roe solver shows better contact capturing ability and mass conservation. [ABSTRACT FROM AUTHOR]

Partial discharges and the electrical treeing represent one of key aspects for the ageing of the insulating parts of electrical components. To predict the end of life of dielectric systems several different modelling approaches have been developed. One of the most promising is the modelling using first principles and a set of partial differential equations. In this field, one of the toughest challenges is to cope with the very different scales of the treeing (each branch is a few micrometers wide) and of electrical components (raging from a few centimetres to a few meters). In this paper we propose an approach, based on the finite element and finite volume methods, which uses two completely independent grids to describe internal defects and electrical components. Some interpolation algorithms have been developed to couple the solution on the two grids. We have also tested the accuracy of our approach by comparing it with previously published methods that use only one grid. • Partial discharges simulation using plasma physics for challenging geometries. • Efficient treatment of tiny internal defects in electric equipments. • Grid decoupling between a gaseous defect and a dielectric bulk. [ABSTRACT FROM AUTHOR]

The solution of Poisson-like equations defined on a complex geometry is required for gyrokinetic simulations, which are important for the modelling of plasma turbulence in nuclear fusion devices such as the ITER tokamak. In this paper, we compare three existing solvers finely tuned to solve this problem, in terms of the accuracy of the solution, and their computational efficiency. We also consider practical implementation aspects, including the parallel efficiency of the code, potentially enabling an integration of the solvers in a state-of-the-art first-principle gyrokinetic simulation framework. The first, the Spline FEM solver, uses C 1 polar splines to construct a finite elements method which solves the equation on curvilinear coordinates. The resulting linear system is solved using a conjugate gradient method. The second, the GMGPolar solver, uses a symmetric finite difference method to discretise the differential equation. The resulting linear system is solved using a tailored geometric multigrid scheme, with a combination of zebra circle and radial line smoothers, together with an implicit extrapolation scheme. The third, the Embedded Boundary solver, uses a finite volumes method on Cartesian coordinates with an embedded boundary scheme. The resulting linear system is solved using a multigrid scheme. The Spline FEM solver is shown to be the most accurate. The GMGPolar solver is shown to use the least memory. The Embedded Boundary solver is shown to be the fastest in most cases. All three solvers are shown to be capable of solving the equation on a realistic non-analytical geometry. The Embedded Boundary solver is additionally used to attempt to solve an X-point geometry. • Solver comparison for the gyrokinetic Poisson-like equation on complex geometry. • High-order spline FEM solver based on C1 polar splines and conjugate gradient. • Curvilinear finite difference method using multigrid with low memory footprint. • Embedded boundary scheme with fast multigrid for simple X-point geometry. [ABSTRACT FROM AUTHOR]

The paper extends to moist-precipitating dynamics a recently documented high-performance finite-volume module (FVM) for simulating global all-scale atmospheric flows (Smolarkiewicz et al., 2016) [62] . The thrust of the paper is a seamless coupling of the conservation laws for moist variables engendered by cloud physics with the semi-implicit, non-oscillatory forward-in-time integrators proven for dry dynamics of FVM. The representation of the water substance and the associated processes in weather and climate models can vary widely in formulation details and complexity levels. The representation adopted for this paper assumes a canonical “warm-rain” bulk microphysics parametrisation, recognised for its minimal physical intricacy while accounting for the essential mathematical complexity of cloud-resolving models. A key feature of the presented numerical approach is global conservation of the water substance to machine precision—implied by the local conservativeness and positivity preservation of the numerics—for all water species including water vapour, cloud water, and precipitation. The moist formulation assumes the compressible Euler equations as default, but includes reduced anelastic equations as an option. The theoretical considerations are illustrated with a benchmark simulation of a tornadic thunderstorm on a reduced size planet, supported with a series of numerical experiments addressing the accuracy of the associated water budget. [ABSTRACT FROM AUTHOR]

In this paper, the second in a series, the high order sub-cell finite volume method is extended to two-dimensional hyperbolic systems on unstructured quadrilateral grids. The basic idea of this method is to subdivide a control volume (main cell) into several sub-cells and the finite volume method is applied to each of the sub-cells. The average values on the sub-cells belonging to current and face neighboring main cells are used to reconstruct a common polynomial distribution of a dependent variable on current main cell. This method can achieve high order accuracy using a compact stencil. The focus of this paper is to study the performance of the sub-cell finite volume method on two-dimensional unstructured quadrilateral grids and to verify that high order accuracy can be achieved using face neighboring sub-cells only. Fourier analysis is performed to analyze the dispersion and dissipation properties of the two-dimensional sub-cell finite volume schemes. To capture the discontinuities, the paper proposes a cell-based multi-dimensional limiting procedure using only face-neighboring main cells. Several benchmark test cases are simulated to validate the proposed sub-cell finite volume schemes and the multi-dimensional limiting procedure. [ABSTRACT FROM AUTHOR]

This paper presents a variational reconstruction for the high order finite volume method in solving the two-dimensional Navier–Stokes equations on arbitrary unstructured grids. In the variational reconstruction, an interfacial jump integration is defined to measure the jumps of the reconstruction polynomial and its spatial derivatives on each cell interface. The system of linear equations to determine the reconstruction polynomials is derived by minimizing the total interfacial jump integration in the computational domain using the variational method. On each control volume, the derived equations are implicit relations between the coefficients of the reconstruction polynomials defined on a compact stencil involving only the current cell and its direct face-neighbors. The reconstruction and time integration coupled iteration method proposed in our previous paper is used to achieve high computational efficiency. A problem-independent shock detector and the WBAP limiter are used to suppress non-physical oscillations in the simulation of flow with discontinuities. The advantages of the finite volume method using the variational reconstruction over the compact least-squares finite volume method proposed in our previous papers are higher accuracy, higher computational efficiency, more flexible boundary treatment and non-singularity of the reconstruction matrix. A number of numerical test cases are solved to verify the accuracy, efficiency and shock-capturing capability of the finite volume method using the variational reconstruction. [ABSTRACT FROM AUTHOR]

In this paper we describe a second order multi-dimensional scheme, belonging to the class of direct Arbitrary Lagrangian–Eulerian (ALE) methods, for the solution of non-linear hyperbolic systems of conservation law. The scheme is constructed upon a cell-centered explicit Lagrangian solver completed with an edge-based upwinded formulation of the numerical fluxes, computed from the MUSCL-Hancock method, to obtain a full ALE formulation. Numerical fluxes depend on nodal grid velocities which are either set or computed to avoid most of the mesh problems typically encountered in purely Lagrangian simulations. In order to assess the robustness of the scheme, most results proposed in this paper have been obtained by computing the grid velocities as a fraction of the Lagrangian nodal velocities, the ratio being set before running the test case. The last part of the paper describes preliminary results about the triple point test case run in the ALE framework by computing the grid velocities with the fully adaptive Large Eddy Limitation (L.E.L.) method proposed in [1] . Such a method automatically computes the grid velocities at each node defining the mesh from the local characteristics of the flow. We eventually discuss the advantages and the drawback of the coupling. [ABSTRACT FROM AUTHOR]

This paper presents a new approach, so-called boundary variation diminishing (BVD), for reconstructions that minimize the discontinuities (jumps) at cell interfaces in Godunov type schemes. It is motivated by the observation that diminishing the jump at the cell boundary can effectively reduce the dissipation in numerical flux. Differently from the existing practices which seek high-order polynomials within mesh cells while assuming discontinuities being always at the cell interfaces, the BVD strategy presented in this paper switches between a high-order polynomial and a jump-like reconstruction that allows a discontinuity being partly represented within the mesh cell rather than at the interface. Excellent numerical results have been obtained for both scalar and Euler conservation laws with substantially improved solution quality in comparison with the existing methods. It is shown that new schemes of high fidelity for both continuous and discontinuous solutions can be devised by the BVD guideline with properly-chosen candidate reconstruction schemes. This work provides a simple and accurate alternative of great practical significance to the current high-order Godunov paradigm which overly pursues the smoothness within mesh cells under the questionable premiss that discontinuities only appear at cell interfaces. [ABSTRACT FROM AUTHOR]

In this paper, we present a pairwise-relaxing meshless (PRM) method for solving the Euler equations of compressible flows within the Arbitrary Lagrangian Eulerian (ALE) framework. Derived from the moving particle semi-implicit (MPS) method and the finite volume particle (FVP) method, the PRM approximates the derivatives from the value defined at the midpoint of each interacting particle pairs through a kernel-based formulation. Pairwise-relaxing constants are introduced to the kernels to provide degree of freedom to enforce the Taylor-series consistency condition while mass, momentum and energy are conserved exactly. An upwind high-order reconstruction scheme via a corrective procedure and variable cut-off radius is also developed for this PRM method. The HLLC approximate Riemann solver is adopted to solve Riemann problem. One and two-dimensional numerical tests are presented to demonstrate the performance of the PRM method. • Second-order accuracy and conservation property are guaranteed in the PRM. • An upwind reconstruction process via a corrective procedure is developed. • Variable cut-off radius is used. [ABSTRACT FROM AUTHOR]

In this paper, we present a new limiter for discontinuous Galerkin (DG) schemes, based on subcell resolution through reconstructed flux correction, for hyperbolic conservation laws. This limitation technique is constructed by means of a subcell Finite Volume (FV) formulation. The algorithm is thus simple, and is able to retain the very high accurate subcell resolution of DG schemes. The main idea of this correction procedure is to preserve as much as possible the high accuracy and the very precise subcell resolution of DG schemes. Consequently, an a posteriori correction will only be applied locally at the subcell scale where it is needed, but still ensuring the scheme conservation. To do so, we first reformulate DG schemes as a subcell FV scheme provided the use of the correct numerical flux. This very simple development provides us with the so-called DG reconstructed flux. This theoretical part will serve as a basis for our limiter framework. Practically, at each time step, we compute a DG candidate solution and check if this solution is admissible (for instance positive, non-oscillating, ...). If it is the case, we go further in time. Otherwise, we return to the previous time step and correct locally, at the subcell scale, the numerical solution. This is why it is refereed to as a posteriori limitation. To this end, each cell is subdivided into subcells. Then, if the solution is locally detected as bad, we substitute the DG reconstructed flux on the subcell boundaries by a robust first-order or second-order TVD numerical flux. And for subcell detected as admissible, we keep the high-order reconstructed flux which allows us to retain the very high accurate resolution and conservation of the DG scheme. Furthermore, only the solution inside troubled subcells and its first neighbors will have to be recomputed, elsewhere the solution remains unchanged. Numerical results on various type problems and test cases will be presented, both in 1D and 2D on Cartesian grids, to assess the very good performance of the design limiting algorithm. • DG schemes can be reformulated as a finite volume method on a subgrid. • This reformulation relies on the definition of the so-called reconstructed flux. • Through its modification, a new a posteriori correction of DG schemes is introduced. • This correction is high-order accurate, positivity-preserving and totally parameter free. • This correction acts at the subcell scale, without impacting the solution elsewhere in the cell. [ABSTRACT FROM AUTHOR]

Abstract We propose a general class of genuinely two-dimensional incomplete Riemann solvers for systems of conservation laws. In particular, extensions of Balsara's multidimensional HLL scheme [3] to two-dimensional PVM/RVM (Polynomial/Rational Viscosity Matrix) finite volume methods are considered. The numerical flux is constructed by assembling, at each edge of the computational mesh, a one-dimensional PVM/RVM flux with two purely two-dimensional PVM/RVM fluxes at vertices, which take into account transversal features of the flow. The proposed methods are applicable to general hyperbolic systems, although in this paper we focus on applications to magnetohydrodynamics. In particular, we propose an efficient technique for divergence cleaning of the magnetic field that provides good results when combined with our two-dimensional solvers. Several numerical tests including genuinely two-dimensional effects are presented to test the performances of the proposed schemes. Highlights • We propose a new class of genuinely two-dimensional incomplete Riemann solvers. • Our solvers can be applied to arbitrary hyperbolic systems. • They can be interpreted as extensions of the HLL 2d scheme proposed by D. Balsara. • Applications to magnetohydrodynamics are presented. • We have also proposed a new technique for divergence cleaning in MHD. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a unified gas-kinetic scheme (UGKS) for multiphase dilute gas-particle system is proposed. The UGKS multiphase (UGKS-M) is a finite volume method, which captures flow physics in the regimes from collisionless multispecies transport to the two-fluid hydrodynamic Navier-Stokes (NS) solution with the variation of Knudsen number, and from granular flow regime to dusty gas dynamics with the variation of Stokes number. The reason for preserving the multiscale nature in UGKS-M is mainly coming from the direct modeling of the flow physics in the scales of discrete cell size and time step, where the ratio of the time step to the particle collision time determines flow behavior in different regimes. For the particle phase, the time evolution solution of the kinetic model equation is used in the construction of numerical flux, which takes into account the particle transport, collision, and acceleration. The gas phase is assumed to be in the continuum flow regime and evolves numerically by the gas-kinetic scheme (GKS), which is a subset of the UGKS for the Navier-Stokes solutions. The interaction between the gas and particle phase is calculated based on a velocity space mapping method, which solves accurately the kinetic acceleration process. The stability of UGKS-M is determined by the CFL condition only. With the inclusion of the material temperature evolution equation of solid particles, once the total energy loss in inelastic collision transfers into particle material thermal energy, the UGKS-M conserves the total mass, momentum, and energy for the whole multiphase system. In the numerical tests, the UGKS-M shows multiscale property in capturing the particle trajectory crossing (PTC), particle wall reflecting phenomena, and vortex-induced segregation of inertial particles under different Stokes numbers. The scheme is also applied to simulate shock induced fluidization problem, where the simulation results agree well with experimental measurements. Highlights • Construction of a multiscale method UGKS-M for gas-particle system. • The particle phase and gas phase are fully coupled through the momentum and energy exchange. • Different flow regimes are well captured with the variation of Knudsen and Stokes numbers. • The scheme is validated through many non-equilibrium test cases. [ABSTRACT FROM AUTHOR]

Abstract This article deals with the thermodynamic equilibrium recovery mechanisms in two-phase flows and their numerical modeling. The two phases, initially at different pressures, temperatures and chemical potentials, are supposed to be driven towards equilibrium conditions by three relaxation processes. First, a mechanical process applies to relax phasic pressures, then a thermal process, to allow the sensible heat transfer between the phases at different temperatures, and, lastly, a chemical process that is responsible for the mass transfer. The two-phase flow model is composed of six partial differential equations with source terms that allow the description of mixtures at full thermodynamic disequilibrium. Its homogeneous portion is hyperbolic and it is solved by a second-order accurate finite volume scheme that uses a HLLC-type approximate Riemann solver. The source terms modeling the relaxation processes are separately integrated as three systems of ordinary differential equations. The main contributions of this paper are: the capability of describing the possibly non-instantaneous time delay of equilibrium recover in a novel way, the equation of state independence of the numerical scheme, and the possibility to take into account the morphology of the flow pattern by using the interfacial area between phases. Highlights • Development of non-instantaneous relaxation techniques for the equilibrium recovery of the two-phase flows. • Equation of State (EoS) independence of the numerical techniques proposed. • Comparison of an accurate EoS to the Stiffened Gas one. • Analysis of the thermal and chemical disequilibrium using the IAPWS-IF97 equation of state. • Validation of the quasi-instantaneous simulations against the Homogeneous Equilibrium Model (HEM). [ABSTRACT FROM AUTHOR]

Abstract In this paper, our aim is to present (1) an embedded fracture model (EFM) for coupled flow and mechanics problem based on the dual continuum approach on the fine grid and (2) an upscaled model for the resulting fine grid equations. The mathematical model is described by the coupled system of equation for displacement, fracture and matrix pressures. For a fine grid approximation, we use the finite volume method for flow problem and finite element method for mechanics. Due to the complexity of fractures, solutions have a variety of scales, and fine grid approximation results in a large discrete system. Our second focus in the construction of the upscaled coarse grid poroelasticity model for fractured media. Our upscaled approach is based on the nonlocal multicontinuum (NLMC) upscaling for coupled flow and mechanics problem, which involves computations of local basis functions via an energy minimization principle. This concept allows a systematic upscaling for processes in the fractured porous media, and provides an effective coarse scale model whose degrees of freedoms have physical meaning. We obtain a fast and accurate solver for the poroelasticity problem on a coarse grid and, at the same time, derive a novel upscaled model. We present numerical results for the two dimensional model problem. Highlights • We have developed an embedded fracture model for coupled flow and mechanics problem on the fine grid. • We have designed an accurate and computationally effective fully coarse grid model using nonlocal multicontinuum upscaling. • We have performed numerical results for constrained energy minimization based upscaling of poroelasticity problem. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a generalised perturbation form of the nonhydrostatic partial differential equations (PDEs) that govern dynamics of all-scale global atmospheric flows. There can be many alternative perturbation forms for any given system of the governing PDEs, depending on the assumed ambient state about which perturbations are taken and on subject preferences in the numerical model design. All such forms are mathematically equivalent, yet they have different implications for the design and accuracy of effective semi-implicit numerical integrators of the governing PDEs. Practical and relevant arguments are presented in favour of perturbation forms that maximise the degree of implicitness of the associated integrators. Two optional forms are implemented in the high-performance finite-volume module (IFS-FVM) for simulating global all-scale atmospheric flows Smolarkiewicz et al. (2016) [32]. Their relative performance is verified with a class of ambient states of reduced complexity. A series of numerical simulations of the planetary baroclinic instability assumes geostrophically balanced zonally uniform ambient flows with significant meridional and vertical shear and exemplifies accuracy gains enabled by the generalised perturbation approach. The newly developed semi-implicit integrators show the potential for numerically accurate separation of a background state from finite-amplitude perturbations of the global atmosphere and provide a base for further development. Highlights • New perturbation equations for of all-scale global atmospheric dynamics are derived. • Novel semi-implicit nonoscillatory forward-in-time solvers are developed for the derived PDEs. • Simulations of global baroclinic instability verify the new equations and their solvers. • Simulations show improved accuracy of the new solvers for extended range forecasting. [ABSTRACT FROM AUTHOR]

Abstract In cylindrical/spherical geometries, MHD solvers using explicit finite-volume methods face restrictive time steps imposed by the CFL condition due to the clustering of cells in the azimuthal direction near the pole/axis. We use a conservative averaging-reconstruction method (Ring Average) on structured cylindrical/spherical mesh to remove this severe time step restriction for multi-dimensional curvilinear finite-volume MHD solvers. The Ring Average technique is implemented as a post-processing step and thus requires no changes to the existing data structure, grid definition or numerical methods in the original MHD solver. This paper describes the Ring Average algorithm and presents simulation results using field loop advection and MHD blast waves in cylindrical/spherical geometries for validation. The algorithm is shown to be inexpensive and easily implemented in existing curvilinear finite-volume MHD solvers. Highlights • Ring Average is a "post-processing" module without modifying the original solver. • Ring Average relaxes time-step restrictions in MHD solvers with axis singularities. • Ring Average conserves mass, momentum and energy for proper shock propagations. • Ring Average works for codes with either divergence cleaning or constrained transport. [ABSTRACT FROM AUTHOR]

Abstract In scalar turbulence it is sometimes the case that the scalar diffusivity is smaller than the viscous diffusivity. The thermally-driven turbulent convection in water is a typical example. In such a case the smallest scale in the problem is the Batchelor scale l b , rather than the Kolmogorov scale l k , as l b = l k / S c 1 / 2 , where Sc is the Schmidt number (or Prandtl number in the case of temperature). In the numerical studies of such scalar turbulence, the conventional approach is to use a single grid for both the velocity and scalar fields. Such single-resolution scheme often over-resolves the velocity field because the resolution requirement for scalar is higher than that for the velocity field, since l b < l k for S c > 1. In this paper we put forward an algorithm that implements the so-called multiple-resolution method with a finite-volume code. In this scheme, the velocity and pressure fields are solved in a coarse grid, while the scalar field is solved in a dense grid. The central idea is to implement the interpolation scheme on the framework of finite-volume to reconstruct the divergence-free velocity from the coarse to the dense grid. We demonstrate our method using a canonical model system of fluid turbulence, the Rayleigh–Bénard convection. We show that, with the tailored mesh design, considerable speed-up for simulating scalar turbulence can be achieved, especially for large Schmidt (Prandtl) numbers. In the same time, sufficient accuracy of the scalar and velocity fields can be achieved by this multiple-resolution scheme. Although our algorithm is demonstrated with a case of an active scalar, it can be readily applied to passive scalar turbulent flows. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in [25,45] , could obtain the optimal order of accuracy in smooth regions, and could simultaneously suppress spurious oscillations near discontinuities. The linear weights of such WENO schemes can be any positive numbers on the condition that their sum equals one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference and finite volume WENO schemes. These new WENO schemes are simple to construct and can be easily implemented to arbitrary high order of accuracy and in higher dimensions. Benchmark examples are given to demonstrate the robustness and good performance of these new WENO schemes. Highlights • A new class of high order finite difference and finite volume WENO schemes are constructed. • These schemes are based on the multi-resolution idea, and a series of unequal-sized hierarchical central spatial stencils. • These schemes can use arbitrary positive linear weights, and are easy to implement for one and multi-dimensions. • These schemes have a gradual degrading of accuracy near discontinuities. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the finite volume method for solving the time-fractional diffusion equation: ∂ t α u − div (A ∇ u) = f , 0 < α < 1. We present and analyze a fully discrete numerical scheme which is based on the linear finite volume method for the spatial discretization and the L 1 difference approximation to ∂ t α u. We first establish a new error bound of O (△ t 1 + δ − α) -order for the L 1 formula under the condition of u (t) ∈ C 1 , δ [ 0 , T ] where δ ∈ [ 0 , 1 ] is the Hölder continuity index. Then, we prove that this fully discrete finite volume scheme is unconditionally stable and the discrete solution admits the optimal error estimate of O (△ t 1 + δ − α + h 2) -order in the L 2 -norm. Numerical examples are provided to support our theoretical analysis. Highlights • We present and analyze a finite difference/finite volume method for the time-fractional diffusion equation: ∂ t α u − div (A ∇ u) = f , 0 < α < 1. • A new error bound of O (△ t 1 + δ − α) -order is established for the L 1 difference formula under the condition of u (t) ∈ C 1 , δ [ 0 , T ] where 0 ≤ δ ≤ 1 is the Hölder index. • We prove that this fully discrete finite volume scheme is unconditionally stable and admits the optimal error estimate of O (△ t 1 + δ − α + h 2) -order in the L 2 norm. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a third-order weighted essentially non-oscillatory (WENO) scheme is developed for the hyperbolic conservation laws on unstructured quadrilateral and triangular meshes. As a starting point, a general stencil is selected for the cell with any local topology, and a unified linear scheme can be constructed. However, in the traditional WENO scheme on unstructured meshes, the very large and negative weights may appear for the mesh with lower local mesh quality, which make the WENO scheme unstable even for the smooth tests. In the current scheme, an optimization approach is given to deal with the very large linear weights, and the splitting technique is considered to deal with the negative weights obtained by the optimization approach. The non-linear weights with the new smooth indicator are proposed as well, in which the local mesh quality and discontinuities of solutions are taken into account simultaneously. Numerical tests are presented to validate the current scheme. The expected convergence rate of accuracy is obtained, and the absolute value of error is not affected by mesh quality. The numerical tests with strong discontinuities validate the robustness of current WENO scheme. Highlights • Third-order WENO schemes are developed on both quadrilateral and triangular meshes. • A general stencil is selected for the cell with any local topology, and a unified linear scheme can be constructed. • An optimization approach is introduced for the very large linear weights, which may appear for the mesh with lower quality. • A new non-linear weight is proposed, where the mesh quality and discontinuities of solutions are considered. [ABSTRACT FROM AUTHOR]

Abstract The paper discusses high-order finite-volume numerical modeling of drift waves, which is an ubiquitous phenomenon in magnetized plasmas. It is found that some standard discretization methods applied to the conservative form of the governing equations can lead to a numerical instability. A method to stabilize high-order discretization is proposed and demonstrated to work in numerical simulations performed with the fourth-order finite-volume code COGENT. As practical examples, a stable drift-wave solution with adiabatic electrons and the collisionless (universal) drift-wave instability driven by electron kinetic effects are considered. Application of the present analysis to a broader range of computational fluid dynamics systems is discussed. Highlights • High-order finite-volume numerical modeling of drift waves and other incompressible flow systems is discussed. • It is shown that some standard high-order finite-volume discretization methods can lead to a numerical instability. • A method to stabilize a high-order discretization is proposed and is demonstrated to work in COGENT simulations. [ABSTRACT FROM AUTHOR]

We are interested in the approximation of a steady hyperbolic problem. In some cases, the solution can satisfy an additional conservation relation, at least when it is smooth. This is the case of an entropy. In this paper, we show, starting from the discretization of the original PDE, how to construct a scheme that is consistent with the original PDE and the additional conservation relation. Since one interesting example is given by the systems endowed by an entropy, we provide one explicit solution, and show that the accuracy of the new scheme is at most degraded by one order. In the case of a discontinuous Galerkin scheme and a Residual distribution scheme, we show how not to degrade the accuracy. This improves the recent results obtained in [1–4] in the sense that no particular constraints are set on quadrature formula and that a priori maximum accuracy can still be achieved. We study the behavior of the method on a non linear scalar problem. However, the method is not restricted to scalar problems. [ABSTRACT FROM AUTHOR]

This paper serves to treat boundary conditions numerically with high order accuracy in order to suit the two-stage fourth-order finite volume schemes for hyperbolic problems developed in J. Li and Z. Du (2016) [17] . As such, it is significant when capturing small scale structures near physical boundaries. Different from previous contributions in literature, the current approach constructs a fourth order accurate approximation to boundary conditions by only using the Jacobian matrix of the flux function (characteristic information) instead of its successive differentiation of governing equations leading to tensors of high ranks in the inverse Lax–Wendroff method. Technically, data in several ghost cells are constructed with interpolation so that the interior scheme can be implemented over boundary cells, and theoretical boundary condition has to be modified properly at intermediate stages so as to make the two-stage scheme over boundary cells fully consistent with that over interior cells. This is nonintuitive and highlights the fact that theoretical boundary conditions are only prescribed for continuous partial differential equations (PDEs), while they must be approximated in a consistent way (even though they could be exactly valued) when the PDEs are discretized . Several numerical examples are provided to illustrate the performance of the current approach when dealing with general boundary conditions. [ABSTRACT FROM AUTHOR]

Numerical discretizations of the Generalized Porous Medium Equation (GPME) with discontinuous coefficients are analyzed with respect to the formation of numerical artifacts. In addition to the degeneracy and self-sharpening of the GPME with continuous coefficients, detailed in [1] , increased numerical challenges occur in the discontinuous coefficients case. These numerical challenges manifest themselves in spurious temporal oscillations in second order finite volume discretizations with both arithmetic and harmonic averaging. The integral average, developed in [2] , leads to improved solutions with monotone and reduced amplitude temporal oscillations. In this paper, we propose a new method called the Shock-Based Averaging Method (SAM) that incorporates the shock position into the numerical scheme. The shock position is numerically calculated by discretizing the theoretical speed of the front from the GPME theory. The speed satisfies the jump condition for integral conservation laws. SAM results in a non-oscillatory temporal profile, producing physically valid numerical results. We use SAM to demonstrate that the choice of averaging alone is not the cause of the oscillations, and that the shock position must be a part of the numerical scheme to avoid the artifacts. [ABSTRACT FROM AUTHOR]