Cell-centered Scheme Research Articles

Two linear finite difference schemes based on exponential basis for two-dimensional time fractional Burgers equation

In present paper, we propose tailored finite point method (TFPM) based on exponential basis for solving two-dimensional time fractional Burgers equation. We use the L1 and L1-2 discretization formulas for the Caputo fractional derivative in time direction and use the TFPM to approximate the spatial derivatives. This method with the help of exponential functions fit the properties of the local solution in time and space simultaneously which act as basis functions in the frame of TFPM. We focus on constructing two linear implicit finite difference schemes, i.e. a five-point node centered scheme and a four-point cell-centered scheme based on exponential functions for deriving the numerical solution of given problem. The stability and convergence of fully discrete schemes based on five-point node centered are discussed. At last, we use some numerical examples to show the accuracy and efficiency of the presented schemes.

A nonlinear correction scheme for the heterogeneous and anisotropic diffusion problems on polygonal meshes

In this paper, a nonlinear positivity-preserving finite volume scheme for the heterogeneous and anisotropic diffusion problems is proposed. Firstly, a linear diamond finite volume scheme is introduced. This scheme has both cell-centered unknowns and vertex unknowns. The vertex unknowns are treated as auxiliary ones and are expressed as linear combinations of the surrounding cell-centered unknowns, which reduces the scheme to a pure cell-centered one. Two new interpolation algorithms are suggested through the linearity-preserving approach and a novel discretization of diffusion coefficient. The new vertex interpolation algorithms improve a great deal the numerical performance of the linear scheme on distorted meshes with strongly anisotropy. Secondly, a new positivity-preserving cell-centered scheme is constructed via the aforementioned linear scheme and a nonlinear correction technique proposed by [28]. This nonlinear scheme is different from most existing nonlinear two-point flux approximation schemes in the way that it is not based on the nonlinear two-point flux approximation and it doesn't require the positivity-preserving vertex interpolation. Its implementation is also very easy and simple. Numerical experiments conducted in this paper indicate that the new positivity-preserving scheme is efficient and has approximately second order accuracy in most extreme cases.

A vertex-centered finite volume scheme preserving the discrete maximum principle for anisotropic and discontinuous diffusion equations

In this paper, we propose one nonlinear vertex-centered finite volume (FV) scheme preserving the discrete maximum principle (DMP) for diffusion equations on distorted mildly meshes. We construct the vertex-centered scheme on dual meshes. By using harmonic average, we get the conservation of scheme. To satisfy the DMP, some existing schemes have to impose severe restrictions on the diffusion coefficient and meshes, which include that only one discontinuity is allowed in the domain. Our scheme removes these restrictions and can deal with the problem with any discontinuity on primary cell edges. Besides, our vertex-centered scheme does not involve the harmonic average of diffusion coefficients in different cells, which is remarkably distinguishable from cell-centered schemes, hence it does not suffer the numerical heat-barrier issue. Numerical results show that our scheme obtains second-order accuracy for continuous problems and those problems with more than one discontinuity. Moreover, our scheme can obtain much better accuracy and computational efficiency than other existing DMP-preserving schemes. At the same time, our scheme can handle the anisotropic problems and satisfies the DMP.

A wide-template and high-accuracy data transfer method for unstructured adjoint-based grid adaptation

Grid adaptation is an important way to improve the identification of flow field and the accuracy of aerodynamic characteristics. Adjoint-based error estimation and grid adaptation method focuses on improving aerodynamic characteristics by optimizing local grids areas which are more sensitive to output functions. The data transfer between different meshes during the iteration of adjoint-based grid adaptation is the basis of accurate error estimation and grid adaptation. The traditional vertex-based method, however, is not suitable for cell-centred scheme finite volume method. A new wide-template and high-accuracy data transfer method for unstructured adjoint-based grid adaptation is proposed in this paper. Numerical results show that the proposed method is more accurate than traditional node-based method when transferring discontinuous flow field containing shock waves in cell-centred scheme finite volume method, and the accuracy of aerodynamic characteristics is obviously improved after grid adaptation.

An Improved Linearity-Preserving Cell-Centered Scheme for Nonlinear Diffusion Problems on General Meshes

An Improved Linearity-Preserving Cell-Centered Scheme for Nonlinear Diffusion Problems on General Meshes

Parallel finite-volume discrete Boltzmann method for inviscid compressible flows on unstructured grids.

In this paper, a finite-volume discrete Boltzmann method based on a cell-centered scheme for inviscid compressible flows on unstructured grids is presented. In the new method, the equilibrium distribution functions are obtained from the circle function in two-dimensions (2D) and the spherical function in three-dimensions (3D). Moreover, the advective fluxes are evaluated by Roe's flux-difference splitting scheme, the gradients of the density and total energy distribution functions are computed with a least-squares method, and the Venkatakrishnan limiter is employed to prevent oscillations. To parallelize the method we use a graph-based partitioning approach that also guarantees the load balancing. The method is validated by seven benchmark problems: (a) a 2D flow pasting a bump, (b) a 2D Riemann problem, (c) a 2D flow passing the RAE2822 airfoil, (d) flows passing the NACA0012 airfoil, (e) 2D supersonic flows around a cylinder, (f) an explosion in a 3D box, and (g) a 3D flow around the ONERA M6 wing. The benchmark tests show that the results obtained by the proposed method match well with the published results, and the parallel numerical experiments show that the proposed parallel implementation has close to linear strong scalability, and parallel efficiencies of 95.31% and 94.56% are achieved for 2D and 3D problems on a supercomputer with up to 4800 processor cores, respectively.

CELL-CENTERED LAGRANGIAN LAX-WENDROFF HLL HYBRID SCHEME ON UNSTRUCTURED MESHES

We have recently introduced a new cell-centered Lax-Wendroff HLL hybrid scheme for Lagrangian hydrodynamics [Fridrich et al. J. Comp. Phys. 326 (2016) 878-892] with results presented only on logical rectangular quadrilateral meshes. In this study we present an improved version on unstructured meshes, including uniform triangular and hexagonal meshes and non-uniform triangular and polygonal meshes. The performance of the scheme is verified on Noh and Sedov problems and its second-order convergence is verified on a smooth expansion test.Finally the choice of the scalar parameter controlling the amount of added artificial dissipation is studied.

A finite volume scheme preserving maximum principle with cell-centered and vertex unknowns for diffusion equations on distorted meshes

In this paper, we propose a new nonlinear finite volume (FV) scheme preserving the discrete maximum principle (DMP) for diffusion equations on distorted meshes. We introduce both cell-centered and vertex unknowns as primary ones. It is well-known that some restrictions on the diffusion coefficients and meshes have to be imposed for existing cell-centered schemes to preserve the DMP (see, e.g., Sheng and Yuan 2011,2018), while our scheme here removes these restrictions and can apply to diffusion problems with arbitrary discontinuity on general meshes. That is, the new scheme avoids failure of handling arbitrary discontinuity on general meshes for existing cell-centered schemes, moreover it is proved that our scheme satisfies the DMP. Numerical results show that our scheme can obtain second-order accuracy, deal with the problems with anisotropic and heterogeneous diffusion coefficient and satisfy the DMP. In addition, it is verified that our scheme can compete with some existing schemes, even behaves better in regard to numerical accuracy and efficiency.

A nonlinear convex combination in the construction of finite volume scheme satisfying maximum principle

In the construction of cell-centered finite volume schemes for general diffusion equations with discontinuous coefficients on distorted meshes, auxiliary unknowns are usually introduced to resolve discontinuities across cell-edges. In general, it is difficult to express unconditionally auxiliary unknowns as a convex combination of primary unknowns around. As we know, all existing methods have to impose certain restrictive conditions on cell-distortion and coefficient discontinuities, especially when designing discrete schemes satisfying the maximum principle. In this paper, we present a nonlinear convex combination method, in which each vertex unknown is eliminated by a nonlinear convex combination of two cell-centered unknowns, which are respectively the maximum and minimum of those cell-centered unknowns around the vertex. As two application examples we present two cell-centered schemes satisfying the maximum principle based on the nonlinear convex combination. An analysis for the discrete flux shows that our scheme can exactly recover the linear solution. Numerical results are presented to show the accuracy of the resulting schemes and verify the discrete maximum principle.

Optimized Schwarz and finite element cell-centered method for heterogeneous anisotropic diffusion problems

The paper is concerned with the derivation and analysis of the optimized Schwarz type method for heterogeneous, anisotropic diffusion problems discretized by the finite element cell-centered (FECC) scheme. Differently from the standard finite element method (FEM), the FECC method involves only cell unknowns and satisfies local conservation of fluxes by using a technique of dual mesh and multipoint flux approximations to construct the discrete gradient operator. Consequently, if the domain is decomposed into nonoverlapping subdomains, the transmission conditions (on the interfaces between subdomains) associated with the FECC scheme are different from those of the standard FEM. We derive discrete Robin-type transmission conditions in the framework of FECC discretization, which include both weak and strong forms of the Robin terms due to the construction of the FECC's discrete gradient operator. Convergence of the associated iterative algorithm for a decomposition into strip-shaped subdomains is rigorously proved. Two dimensional numerical results for both isotropic and anisotropic diffusion tensors with large jumps in the coefficients are presented to illustrate the performance of the proposed methods with optimized Robin parameters.

Surface tension for compressible fluids in ALE framework

We describe an Arbitrary-Lagrangian-Eulerian (ALE) method for the compressible Euler system with capillary force. The algorithm is split in two steps. First, the Lagrangian step is based on cell-centred schemes [9,20,46]. The surface tension force is discretized in order to exactly verify the Laplace law at the discrete level. We also provide a second-order spatial extension and a low-Mach correction, which do not break the well-balanced property of the scheme. The Lagrangian scheme is assessed on several problems, particularly on a linear Richtmyer-Meshkov instability which is the targeted application. The second step is the rezoning and remapping done thanks to a swept-region method using exact intersections near the interface. We use a Volume Of Fluid (VOF) method to track the interface. We describe the treatment of mixed-cells, and in particular the thermodynamics closure and the curvature calculation. The new scheme is used to investigate the influence of surface tension on a non-linear Richtmyer-Meshkov instability.

High-order magnetohydrodynamics for astrophysics with an adaptive mesh refinement discontinuous Galerkin scheme

Modern astrophysical simulations aim to accurately model an ever-growing array of physical processes, including the interaction of fluids with magnetic fields, under increasingly stringent performance and scalability requirements driven by present-day trends in computing architectures. Discontinuous Galerkin methods have recently gained some traction in astrophysics, because of their arbitrarily high order and controllable numerical diffusion, combined with attractive characteristics for high performance computing. In this paper, we describe and test our implementation of a discontinuous Galerkin (DG) scheme for ideal magnetohydrodynamics in the AREPO-DG code. Our DG-MHD scheme relies on a modal expansion of the solution on Legendre polynomials inside the cells of an Eulerian octree-based AMR grid. The divergence-free constraint of the magnetic field is enforced using one out of two distinct cell-centred schemes: either a Powell-type scheme based on nonconservative source terms, or a hyperbolic divergence cleaning method. The Powell scheme relies on a basis of locally divergence-free vector polynomials inside each cell to represent the magnetic field. Limiting prescriptions are implemented to ensure non-oscillatory and positive solutions. We show that the resulting scheme is accurate and robust: it can achieve high-order and low numerical diffusion, as well as accurately capture strong MHD shocks. In addition, we show that our scheme exhibits a number of attractive properties for astrophysical simulations, such as lower advection errors and better Galilean invariance at reduced resolution, together with more accurate capturing of barely resolved flow features. We discuss the prospects of our implementation, and DG methods in general, for scalable astrophysical simulations.

A low-Mach correction for multi-dimensional finite volume shock capturing schemes with application in lagrangian frame

We propose a low-Mach correction for cell-centered schemes in Lagrangian frame. After transposing some classical results from the Eulerian frame to the Lagrangian frame, we show why classical cell-centered Lagrangian schemes are not able to capture the low-Mach regime except by using unreasonably fine meshes. Consequently, we propose a slight modification of the original scheme, which is easy to implement in any scheme using an acoustic Godunov solver on unstructured mesh, and has a negligible cost in term of CPU time. We demonstrate that this modification cures this flaw. The properties of the original semi-discrete scheme (consistency, conservation) are preserved. Particular attention is paid to the entropy condition, proving its compatibility with the proposed modification. We assess this new scheme on several low and high-Mach problems, to demonstrate its good behavior in all regimes. Our last test problem is devoted to the study of the growth rate of instability in convergent configurations. It shows that even if the problem is globally very compressible, the low-Mach correction can have a significant impact on the solution.

A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes

We suggest a new positivity-preserving discrete duality finite volume (DDFV) scheme for anisotropic diffusion problems on general polygonal meshes. Like existing DDFV schemes, this scheme is built on the primary and dual meshes and has finite volume (FV) equations for both vertex and cell-centered unknowns. The transpose of the coefficient matrix for the FV equations of the cell-centered unknowns is an M-matrix while that for the vertex unknowns is not an M-matrix but a symmetric and positive definite matrix. By employing a certain truncation technique for the vertex unknowns, the positivity-preserving property for both categories of unknowns is guaranteed. Local conservation is strictly maintained for the cell-centered unknowns and conditionally maintained for the vertex unknowns. Since the FV equations of the vertex unknowns can be solved independently, the two sets of FV equations are decoupled. In contrast to existing nonlinear positivity-preserving schemes, the new scheme requires no nonlinear iterations for linear problems. For nonlinear problems, the positivity-preserving mechanism of the new scheme is decoupled from its nonlinear iteration so that any nonlinear solver can be adopted. Moreover, the positivity of the discrete solution is proved and the well-posedness is analyzed rigorously for linear problems. In numerical experiments, the new scheme is examined extensively and compared with two positivity-preserving cell-centered schemes and a nonlinear DDFV scheme. Numerical results show that the present DDFV scheme achieves second-order accuracy and preserves the positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids. The high efficiency of the scheme is also demonstrated by the comparison of computation time and number of nonlinear iterations.

High order cell-centered scheme totally based on cell average

This work clarifies the concept of cell average by pointing out the differences between cell average and cell centroid value, which are averaged cell-centered value and pointwise cell-centered value, respectively. Interpolation based on cell averages is constructed and high order QUICK-like numerical scheme is designed for such interpolation. A new approach of error analysis is introduced in this work, which is similar to Taylor’s expansion.

High Order Well-Balanced Weighted Compact Nonlinear Schemes for the Gas Dynamic Equations under Gravitational Fields

AbstractIn this study, we propose a high order well-balanced weighted compact nonlinear (WCN) scheme for the gas dynamic equations under gravitational fields. The proposed scheme is an extension of the high order WCN schemes developed in (S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321). For the purpose of maintaining the exact steady state solution, the well-balanced technique in (Y. Xing, C.-W Shu, J. Sci. Comput. 54 (2013) 645-662) is employed to split the source term into two terms. The proposed scheme can maintain the isothermal equilibrium solution exactly, genuine high order accuracy and resolve small perturbations of the hydrostatic balance state on the coarse meshes. Furthermore, in order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact schemes with local characteristic projections are used to construct different WCN schemes. Several representative one- and two-dimensional examples are simulated to demonstrate the good performance of the proposed schemes.

Cell-centered high-order hyperbolic finite volume method for diffusion equation on unstructured grids

We apply a hyperbolic cell-centered finite volume method to solve a steady diffusion equation on unstructured meshes. This method, originally proposed by Nishikawa using a node-centered finite volume method, reformulates the elliptic nature of viscous fluxes into a set of augmented equations that makes the entire system hyperbolic. We introduce an efficient and accurate solution strategy for the cell-centered finite volume method. To obtain high-order accuracy for both solution and gradient variables, we use a successive order solution reconstruction: constant, linear, and quadratic (k-exact) reconstruction with an efficient reconstruction stencil, a so-called wrapping stencil. By the virtue of the cell-centered scheme, the source term evaluation was greatly simplified regardless of the solution order. For uniform schemes, we obtain the same order of accuracy, i.e., first, second, and third orders, for both the solution and its gradient variables. For hybrid schemes, recycling the gradient variable information for solution variable reconstruction makes one order of additional accuracy, i.e., second, third, and fourth orders, possible for the solution variable with less computational work than needed for uniform schemes. In general, the hyperbolic method can be an effective solution technique for diffusion problems, but instability is also observed for the discontinuous diffusion coefficient cases, which brings necessity for further investigation about the monotonicity preserving hyperbolic diffusion method.

A new nodal solver for the two dimensional Lagrangian hydrodynamics

We describe a cell-centered Godunov type scheme for the Lagrangian hydrodynamic equations on general unstructured meshes with nodal fluxes. The nodal solver only depends on the angular repartition of the physical variables around the node and not on the length of the edges. The scheme verifies a weak consistency property. Numerical results are compared to EUCCLHYD and GLACE schemes which are also cell-centered schemes with node based fluxes for Lagrangian hydrodynamics.

A third-order moving mesh cell-centered scheme for one-dimensional elastic-plastic flows

A third-order moving mesh cell-centered scheme without the remapping of physical variables is developed for the numerical solution of one-dimensional elastic-plastic flows with the Mie–Grüneisen equation of state, the Wilkins constitutive model, and the von Mises yielding criterion. The scheme combines the Lagrangian method with the MMPDE moving mesh method and adaptively moves the mesh to better resolve shock and other types of waves while preventing the mesh from crossing and tangling. It can be viewed as a direct arbitrarily Lagrangian–Eulerian method but can also be degenerated to a purely Lagrangian scheme. It treats the relative velocity of the fluid with respect to the mesh as constant in time between time steps, which allows high-order approximation of free boundaries. A time dependent scaling is used in the monitor function to avoid possible sudden movement of the mesh points due to the creation or diminishing of shock and rarefaction waves or the steepening of those waves. A two-rarefaction Riemann solver with elastic waves is employed to compute the Godunov values of the density, pressure, velocity, and deviatoric stress at cell interfaces. Numerical results are presented for three examples. The third-order convergence of the scheme and its ability to concentrate mesh points around shock and elastic rarefaction waves are demonstrated. The obtained numerical results are in good agreement with those in literature. The new scheme is also shown to be more accurate in resolving shock and rarefaction waves than an existing third-order cell-centered Lagrangian scheme.

Decoupling mixed finite elements on hierarchical triangular grids for parabolic problems

In this paper, we propose a numerical method for the solution of time-dependent flow problems in mixed form. Such problems can be efficiently approximated on hierarchical grids, obtained from an unstructured coarse triangulation by using a regular refinement process inside each of the initial coarse elements. If these elements are considered as subdomains, we can formulate a non-overlapping domain decomposition method based on the lowest-order Raviart–Thomas elements, properly enhanced with Lagrange multipliers on the boundaries of each subdomain (excluding the Dirichlet edges). A suitable choice of mixed finite element spaces and quadrature rules yields a cell-centered scheme for the pressures with a local 10-point stencil. The resulting system of differential-algebraic equations is integrated in time by the Crank–Nicolson method, which is known to be a stiffly accurate scheme. As a result, we obtain independent subdomain linear systems that can be solved in parallel. The behavior of the algorithm is illustrated on a variety of numerical experiments.