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This paper first explores Tadmor’s minimum entropy principle for the special relativistic hydrodynamics (RHD) equations and incorporates this principle into the design of robust high-order discontinuous Galerkin (DG) and finite volume schemes for RHD on general meshes. The proposed schemes are rigorously proven to preserve numerical solutions in a global invariant region constituted by all the known intrinsic constraints: minimum entropy principle, the subluminal constraint on fluid velocity, the positivity of pressure, and the positivity of rest-mass density. Relativistic effects lead to some essential difficulties in the present study which are not encountered in the nonrelativistic case. Most notably, in the RHD case the specific entropy is a highly nonlinear implicit function of the conservative variables, and, moreover, there is also no explicit formula of the flux in terms of the conservative variables. In order to overcome the resulting challenges, we first propose a novel equivalent form of the invariant region by skillfully introducing two auxiliary variables. As a notable feature, all the constraints in the novel form are explicit and linear with respect to the conservative variables. This provides a highly effective approach to theoretically analyze the invariant-region-preserving (IRP) property of numerical schemes for RHD, without any assumption on the IRP property of the exact Riemann solver. Based on this, we prove the convexity of the invariant region and establish the generalized Lax–Friedrichs splitting properties via technical estimates, laying the foundation for our rigorous IRP analyses. It is rigorously shown that the first-order Lax–Friedrichs type scheme for the RHD equations satisfies a local minimum entropy principle and is IRP under a CFL condition. Provably IRP high-order accurate DG and finite volume methods are then developed for the RHD with the help of a simple scaling limiter, which is designed by following the bound-preserving type limiters in the literature. Several numerical examples demonstrate the effectiveness and robustness of the proposed schemes. [ABSTRACT FROM AUTHOR]

The shallow water equations provide the basic modeling equations for a number of coastal flooding hazards, such as tsunamis and storm surge. In realistic scenarios, there are often structures important to these flows that have a large extent but small width, including sea walls, berms, and harbor barriers. Explicit time stepping schemes, most often used for the shallow water equations, can then suffer from time step restrictions due to the CFL condition. In this article, we introduce an approach that side-steps these issues by allowing a barrier to have zero-width and to cut a cell arbitrarily without suffering from CFL restrictions. This is done by supplementing existing Riemann solvers and leveraging h-box and large time stepping methods. These methods preserve the properties of the Riemann solver and add negligible cost to the original solvers. [ABSTRACT FROM AUTHOR]

We study the Lax consistency of a staggered finite volume scheme for Lagrangian hydrodynamics adapting the methodology introduced by Despres in [Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 2669-2679]. The scheme is conservative in mass, total energy, and momentum, and the positivity of the density and internal energy are ensured under a CFL like condition. In addition, the scheme is weakly consistent on general meshes in any dimension. The key idea of the proof is the discrete div-grad duality property derived from the discretization of the operators. [ABSTRACT FROM AUTHOR]

This paper extends the author's previous two-dimensional work with Ou and Le-Veque to high-resolution finite volume modeling of systems of fluids and poroelastic media in three dimensions, using logically rectangular mapped grids. A method is described for calculating consistent cell face areas and normal vectors for a finite volume method on a general nonrectilinear hexahedral grid. A novel limiting algorithm is also developed to cope with difficulties encountered in implementing high-resolution finite volume methods for anisotropic media on nonrectilinear grids; the new limiting approach is compatible with any limiter function and typically reduces solution error even in situations where it is not necessary for correct functioning of the numerical method. Dimensional splitting is used to reduce the computational cost of the solution. The code implementing the three-dimensional algorithms is verified against known plane wave solutions, with particular attention to the performance of the new limiter algorithm in comparison to the classical one. An acoustic wave in brine striking an uneven bed of orthotropic layered sandstone is also simulated in order to demonstrate the capabilities of the simulation code. [ABSTRACT FROM AUTHOR]

We present the multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations. Given a system of differential equations possessing conservation laws, our approach is based on discretizing conservation law multipliers and their associated density and flux functions. We show that the proposed discretization is consistent for any order of accuracy when the discrete multiplier has a multiplicative inverse. Moreover, we show that by construction, discrete densities can be exactly conserved. In particular, the multiplier method does not require the system to possess a Hamiltonian or variational structure. Examples, including dissipative problems, are given to illustrate the method. In the case when the inverse of the discrete multiplier becomes singular, consistency of the method is also established for scalar ODEs provided the discrete multiplier and density are zero compatible. [ABSTRACT FROM AUTHOR]

We present a conservative method for the three-dimensional coupling between an inviscid compressible flow and a moving rigid solid. We consider an inviscid Euler fluid in conservative form discretized using a high-order monotonicity-preserving finite volume method with a directional operator splitting. An immersed boundary technique is employed through the modification of the finite volume fluxes in the vicinity of the solid. The method yields exact conservation of mass, momentum, and energy of the system, and also exhibits important consistency properties, such as conservation of uniform movement of both fluid and solid as well as the absence of numerical roughness on a straight boundary. The coupling scheme evaluates the fluxes on the fluid side and the forces and torques on the solid side only once every time step, ensuring the computational efficiency of the coupling. We present numerical results assessing the robustness of the method in the case of rigid solids with large displacements. [ABSTRACT FROM AUTHOR]

We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar diffusion equation. The current analysis deviates significantly from the previous in three respects. First, additional stabilization leads to a more complex saddle-point problem. Second, a discrete Korn's inequality has to be established for the global discretization. Finally, robustness with respect to the Poisson ratio is analyzed. The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity. [ABSTRACT FROM AUTHOR]

We present a novel well-balanced second order Godunov-type finite volume scheme for compressible Euler equations with gravity. The well-balanced property is achieved by a specific combination of source term discretization, hydrostatic reconstruction, and numerical flux that exactly resolves stationary contacts. The scheme is able to preserve isothermal and polytropic stationary solutions up to machine precision. It is applied on several examples using the numerical flux of Roe to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution. [ABSTRACT FROM AUTHOR]

In this paper, we propose some new coarse correction matrices to design domain decomposition (DD) preconditioners for solving a multiscale finite volume algebraic system. The key ingredients of our coarse correction matrices are based on several operators: prolongation, restriction, and correction operators. Using the coarse correction matrices, together with a one-level additive Schwarz method as a local solver, we may get several efficient preconditioners for the fine-scale finite volume linear system. Techniques used in some well-known multiscale methods are important and inspire us to combine them to generate different kinds of operators. It is shown that our new coarse correction matrices are more robust than well-known ones in the literature. In addition, mass conservation on a primal coarse grid is preserved by a postprocessing procedure, so a conservative fine velocity field may be reconstructed in any interesting local domain. A variety of numerical examples are presented to confirm the validity and robustness of our coarse correction matrices. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the numerical approximation of the classical time-dependent drift-diffusion system near quasi-neutrality. We consider a fully implicit in time and finite volume in space scheme, where the convection-diffusion fluxes are approximated by Scharfetter-Gummel fluxes. We establish that all the a priori estimates needed to prove the convergence of the scheme do not depend on the Debye length λ. This proves that the scheme is asymptotic preserving in the quasi-neutral limit λ → 0. [ABSTRACT FROM AUTHOR]

In this work we develop a high-resolution mapped-grid finite volume method code to model wave propagation in two dimensions in systems of multiple orthotropic poroelastic media and/or fluids, with curved interfaces between different media. We use a unified formulation to simplify modeling of the various interface conditions--open pores, imperfect hydraulic contact, or sealed pores--that may exist between such media. Our numerical code is based on the clawpack framework, but in order to obtain correct results at a material interface we use a modified transverse Riemann solution scheme, and at such interfaces are forced to drop the second-order correction term typical of high-resolution finite volume methods. We verify our code against analytical solutions for reflection and transmission of waves at a material interface, and for scattering of an acoustic wave train around an isotropic poroelastic cylinder. For reflection and transmission at a flat interface, we achieve second-order convergence in the 1-norm and first-order in the max-norm; for the cylindrical scatterer, the highly distorted grid mapping degrades performance but we still achieve convergence at a reduced rate. We also simulate an acoustic pulse striking a simplified model of a human femur bone as an example of the capabilities of the code. [ABSTRACT FROM AUTHOR]

We present a novel mesh-free scheme for solving partial differential equations. We first derive a conservative and stable formulation of mesh-free first derivatives. We then show that this formulation is a special case of a general conservative mesh-free framework that allows flexible choices of flux schemes. Necessary conditions and algorithms for calculating the coefficients for our mesh-free schemes that satisfy these conditions are also discussed. We include numerical examples of solving the one- and two-dimensional inviscid advection equations, demonstrating the stability and convergence of our scheme and the potential of using the general mesh-free framework to extend finite volume discretization to a mesh-free context. [ABSTRACT FROM AUTHOR]

We prove first order convergence of the flux and the pressure for the multipoint flux approximation (MPFA) method known as the MPFA L-method. The proof is based on that presented for the mimetic finite difference paper by Brezzi, Lipnikov, and Shashkov in 2005. Particular to this procedure is that no velocity interpolation is needed inside the element. Hence the volume integration of the velocity field is substituted with a quadrature involving only the fluxes. The proof is valid on polyhedra but limits the possibility of the MPFA L-method to choose a stencil to each and every half-face, as is the usual procedure. The assumption needed for convergence is local and poses limits on grid deformation and the anisotropy of the permeability tensor. This assumption is then used to compare the convergence properties of the MPFA L-method with those of the MPFA O-method and the mimetic finite difference method. [ABSTRACT FROM AUTHOR]

This paper establishes the equivalence of the conforming Courant finite element method, the nonconforming Crouzeix--Raviart finite element method, and several first-order discontinuous Galerkin finite element methods in the sense that the respective energy error norms are equivalent up to generic constants and higher-order data oscillations in a Poisson model problem. The Raviart--Thomas mixed finite element method is better than the previous methods, whereas the conjecture of the converse relation is proved to be false. This paper completes the analysis of comparison initiated by Braess [Calcolo, 46 (2009), pp. 149-155]. Two numerical benchmarks illustrate the comparison theorems and the possible strict superiority of the Raviart--Thomas mixed finite element method. Applications include least-squares finite element methods, finite volume methods, and equality of approximation classes for concepts of optimality for adaptive finite element methods. [ABSTRACT FROM AUTHOR]

We propose a second order finite volume scheme for nonlinear degenerate parabolic equations which admit an entropy functional. For some of these models (porous media equation, drift-diffusion system for semiconductors) it has been proved that the transient solution converges to a steady state when time goes to infinity. The present scheme preserves steady states and provides a satisfying long time behavior. Moreover, it remains valid and second order accurate in space even in the degenerate case. After describing the numerical scheme, we present several numerical results which confirm the high order accuracy in various regime degenerate and nondegenerate cases and underline the efficiency to preserve the large time asymptotic. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of a weakly imposed boundary condition for the one-dimensional Euler equations of gasdynamics. We give a mathematical proof of the convergence in two specific cases, and we analyze two kinds of physically relevant boundary conditions including the nonpenetration condition and the imposed pressure condition. We also show the results of a couple of one-dimensional test cases that confirm our theoretical estimates. [ABSTRACT FROM AUTHOR]

Novel cell-centered full-tensor finite-volume methods are presented for general unstructured grids in two spatial dimensions. The numerical schemes are flux-continuous and based on computing the transmissibilities in a local subcell transform space, ensuring that local flux matrices are symmetric. As a result the global discretization matrix is shown to be symmetric positive definite for any grid type. A symmetric physical space method is also introduced, and the symmetric methods are shown to be closely related. Discrete ellipticity conditions are derived for positive definiteness of the physical space and subcell space schemes. Computational examples are presented for unstructured triangular grids demonstrating good performance of the scheme. The schemes are compared with the so-called multipoint flux approximation (MPFA) O-method [I. Aavatsmark, T. Barkve, Ø. Bøe, and T. Mannseth, SIAM J. Sci. Comput., 19 (1998), pp. 1700-1716]. Good agreement between the methods is obtained, but the new scheme shows improved behavior in challenging cases. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a new kind of meshfree scheme based on a new class of meshfree derivatives: the renormalized meshfree derivatives, which improve the consistency of the original weighted particle methods. The weak renormalized meshfree scheme, built from the weak formulation of general conservation laws, turns out to be L2 stable under some geometrical conditions on the distribution of particles and some regularity conditions of the transport field. A time discretization is then performed by analogy with finite volume methods, and the L1, L∞, and BV stabilities of the obtained time discretized scheme are studied. From the same analogy with finite volume methods, a hybrid particle scheme is built using the Godunov method and is numerically compared to the weak renormalized scheme. [ABSTRACT FROM AUTHOR]

Recently, Courbet and Croisille [RAIRO Modél. Math. Anal. Numér., 32 (1998), pp. 631-649] introduced the finite volume box-scheme for the two-dimensional (2D) Poisson problem in the case of triangular meshes. Generalizations to higher degree box-schemes have been published by Croisille and Greff [Numer. Methods Partial Differential Equations, 18 (2002), pp. 355-373]. These box-schemes are based on the principle of the finite volume method in that they take the average of the equations on each cell of the grid. This gives rise to a natural choice of unknowns located at the interface of the mesh. These box-schemes are conservative and use only one mesh. They can be interpreted as a discrete mixed Petrov-Galerkin formulation of the Poisson problem. In this paper we focus our interest on box-schemes for the Poisson problem in two dimensions on rectangular grids. We discuss the basic finite volume box-scheme and analyze and interpret it as three different box-schemes. The method is demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

We study a colocated cell-centered finite volume method for the approximation of the incompressible Navier--Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressure, all of them colocated at the center of the cells of a unique mesh; such a configuration is known to lead to stability problems, hence the need for a stabilization technique, which we choose of the Brezzi--Pitkäranta type. The scheme features two essential properties: the discrete gradient is the transpose of the divergence terms, and the discrete trilinear form associated to nonlinear advective terms vanishes on discrete divergence free velocity fields. As a consequence, the scheme is proved to be unconditionally stable and convergent for the Stokes problem and for the transient and the steady Navier--Stokes equations. In this latter case, for a given sequence of approximate solutions computed on meshes the size of which tends to zero, we prove, up to a subsequence, the L²-convergence of the components of the velocity, and, in the steady case, the weak L²-convergence of the pressure. The proof relies on the study of space and time translates of approximate solutions, which allows the application of Kolmogorov's theorem. The limit of this subsequence is then shown to be a weak solution of the Navier--Stokes equations. Numerical examples are performed to obtain numerical convergence rates in both the linear and nonlinear cases. [ABSTRACT FROM AUTHOR]

We are interested in a robust and accurate domain decomposition method with arbitrary interface conditions on nonmatching multiblock grids using a finite volume discretization. To take into account the nonmatching grids at the interface, we introduce transmission operators, Dirichlet--Neumann interface conditions, and arbitrary equivalent interface conditions (for example, Robin interface conditions). Under a compatibility assumption on the transmission operators, we prove the equivalence between the different types of interface conditions and the well posedness of the local and global problems. Then two error estimates are proven in terms of the discrete H1-norm: the first in O (h) 1/2 with transmission operators based on piecewise constant functions and the second in O (h) (as in the matching case) with transmission operators using a piecewise linear rebuilding. In conclusion, numerical results are presented in confirmation of the theory. [ABSTRACT FROM AUTHOR]

Scalar conservation laws with a flux function discontinuous in space are approximated using a Godunov-type method for which a convergence theorem is proved. The case where the flux functions at the interface intersect is emphasized. A very simple formula is given for the interface flux. A numerical comparison between the Godunov numerical flux and the upstream mobility flux is presented for two-phase flow in porous media. A consequence of the convergence theorem is an existence theorem for the solution of the scalar conservation laws under consideration. Furthermore, for regular solutions, uniqueness has been shown. [ABSTRACT FROM AUTHOR]

We show here the convergence of the finite volume approximate solutions of a convection-diffusion equation to a weak solution, without the usual coercitivity assumption on the elliptic operator and with weak regularity assumptions on the data. Numerical experiments are performed to obtain some rates of convergence in two and three space dimensions. [ABSTRACT FROM AUTHOR]

We consider solving three-dimensional electromagnetic problems in parameter regimes where the quasi-static approximation applies and the permeability, permittivity, and conductivity may vary significantly. The difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative methods for the solution of the linear systems of algebraic equations that arise when discretizing Maxwell's equations in the frequency domain. The present article extends methods we proposed earlier for constant permeability [E. Haber, U. Ascher, D. Aruliah, and D. Oldenburg, J. Comput. Phys., 163 (2000), pp. 150–171; D. Aruliah, U. Ascher, E. Haber, and D. Oldenburg, Math. Models Methods Appl. Sci., to appear.] to handle also problems in which the permeability is variable and may contain significant jump discontinuities. In order to address the problem of slow convergence we reformulate Maxwell's equations in terms of potentials, applying a Helmholtz decomposition to either the electric field or the magnetic field. The null space of the curl operators can then be annihilated by adding a stabilizing term, using a gauge condition, and thus obtaining a strongly elliptic differential operator. A staggered grid finite volume discretization is subsequently applied to the reformulated PDE system. This scheme works well for sources of various types, even in the presence of strong material discontinuities in both conductivity and permeability. The resulting discrete system is amenable to fast convergence of ILU-preconditioned Krylov methods. We test our method using several numerical examples and demonstrate its robust efficiency. We also compare it to the classical Yee method using similar iterative techniques for the resulting algebraic system, and we show that our method is significantly faster, especially for electric sources. [ABSTRACT FROM AUTHOR]

The numerical approximation of combustion processes may lead to numerical difficulties, which are caused by different time scales of the transport part and the reactive part of the model equations. Here we consider a modified fractional step method that overcomes this difficulty on standard test problems and allows the use of a mesh width and time step determined by the nonreactive part, without precisely resolving the very small reaction zone. High-resolution Godunov methods are employed and the structure of the Riemann solution is used to determine where burning should occur in each time step. The modification is implemented in the software package clawpack. Numerical results for 1D and 2D detonation waves are shown, including a detonation wave diffracting around a corner. [ABSTRACT FROM AUTHOR]

Given a P1 conforming or nonconforming Galerkin finite element method (GFEM) solution ph, which approximates the exact solution p of the diffusion-reaction equation - ∇·Κ ∇+ αp = f with full tensor variable coefficient Κ, we evaluate the approximate flux uh to the exact flux u = -Κ ∇p by a simple but physically intuitive formula over each finite element. The flux is sought in the continuous (in normal component) or the discontinuous Raviart­Thomas space. A systematic way of deriving such a formula is introduced. This direct method retains local conservation property at the element level, typical of mixed methods (finite element or finite volume type), but avoids solving an indefinite linear system. In short, the present method retains the best of the GFEM and the mixed method but without their shortcomings. Thus we view our method as a conservative GFEM and demonstrate its equivalence to a certain mixed finite volume box method. The equivalence theorems explain how the pressure can decouple basically cost free from the mixed formulation. The accuracy in the flux is of first order in the H(div;Ω) norm. Numerical results are provided to support the theory. [ABSTRACT FROM AUTHOR]

Finite volume or finite element methods now dominate the modeling of steady flows because of their natural handling of complex configurations. They should have a similar advantage for unsteady flows; however, their error analysis on nonuniform meshes has met a number of difficulties. In this paper a new approach is adopted which makes greater use of a Godunov formulation. It opens the way for a fuller comparison of characteristic-based methods with semidiscrete methods. [ABSTRACT FROM AUTHOR]

This paper initiates a study of finite volume methods for linear first-order elliptic systems by performing a stability and convergence analysis of the cell vertex approximation of the Cauchy­Riemann equations. The approach is based on reformulating the scheme as a Petrov­Galerkin finite element method with continuous bilinear trial functions and piecewise constant test functions. Optimal error bounds are derived in a mesh-dependent norm, and the counting problem which may occur due to geometry and boundary conditions is considered. [ABSTRACT FROM AUTHOR]