Cell-centered Finite Volume Scheme Research Articles

A multi-dimensional, robust, and cell-centered finite-volume scheme for the ideal MHD equations

We present a new multi-dimensional, robust, and cell-centered finite-volume scheme for the ideal MHD equations. This scheme relies on relaxation and splitting techniques and can be easily used at high order. A fully conservative version is not entropy satisfying but is observed experimentally to be more robust than standard constrained transport schemes at low plasma beta. At very low plasma beta and high Alfvén number, we have designed an entropy-satisfying version that is not conservative for the magnetic field but preserves admissible states and we switch locally a-priori between the two versions depending on the regime of plasma beta and Alfvén number. This strategy is robust in a wide range of standard MHD test cases, all performed at second order with a classic MUSCL-Hancock scheme.

The cell-centered positivity-preserving finite volume scheme for 3D convection–diffusion equation on distorted meshes

PurposeThis paper aims to construct positivity-preserving finite volume schemes for the three-dimensional convection–diffusion equation that are applicable to arbitrary polyhedral grids.Design/methodology/approachThe cell vertices are used to define the auxiliary unknowns, and the primary unknowns are defined at cell centers. The diffusion flux is discretized by the classical nonlinear two-point flux approximation. To ensure the fully discrete scheme has positivity-preserving property, an improved discretization method for the convection flux was presented. Besides, a new positivity-preserving vertex interpolation method is derived from the linear reconstruction in the discretization of convection flux. Moreover, the Picard iteration method may have slow convergence in solving the nonlinear system. Thus, the Anderson acceleration of Picard iteration method is used to solve the nonlinear system. A condition number monitor of matrix is employed in the Anderson acceleration method to achieve better robustness.FindingsThe new scheme is applicable to arbitrary polyhedral grids and has a second-order accuracy. The results of numerical experiments also confirm the positivity-preserving of the discretization scheme.Originality/value1. This article presents a new positivity-preserving finite volume scheme for the 3D convection–diffusion equation. 2. The new discretization scheme of convection flux is constructed. 3. A new second-order interpolation algorithm is given to eliminate the auxiliary unknowns in flux expressions. 4. An improved Anderson acceleration method is applied to accelerate the convergence of Picard iterations. 5. This scheme can solve the convection–diffusion equation on the distorted meshes with second-order accuracy.

A mimetic interpolation-free cell-centered finite volume scheme for the 2D and 3D heterogeneous anisotropic diffusion equations

We propose a mimetic interpolation-free cell-centered finite volume scheme for diffusion equations on arbitrary two-dimensional (2D) and three dimensional (3D) unstructured, possibly non star-shaped or nonplanar meshes. This scheme borrows ideas from the well known mimetic finite difference methods. This interpolation-free means that we do not need to introduce auxiliary unknowns in the process of constructing the scheme, and the final scheme has only cell-centered unknowns. After carefully handling continuity of solutions and normal fluxes, we construct piecewise linear approximations on each cell. This makes our scheme applicable to arbitrary discontinuities and arbitrary mesh topologies. Interpolation-free makes the program easy to implement and the 2D and 3D programs have a unified framework. It is worth mentioning that the traditional star-shaped mesh assumption has been abolished, and the new scheme has more relaxed restrictions on meshes. Some representative and even challenging numerical examples verify that the scheme is robust and almost has the optimal convergence order on some benchmark polygonal and polyhedral meshes.

An unstructured preconditioned central difference finite volume multiphase Euler solver for computing inviscid cavitating flows over arbitrary two- and three-dimensional geometries

In the present work, a numerical method is adopted and applied for simulating the inviscid cavitating flows around two- and three-dimensional geometries on unstructured meshes. The algorithm uses the preconditioned multiphase Euler equations discretized by a cell-centered central difference finite volume scheme with suitable dissipation terms. The interface capturing method with three transport equation-based cavitation models, namely the Merkle et al., Singhal et al. and Kunz et al. models are employed for the mass transfer between the liquid and vapor phases to be calculated. The simulations of the steady inviscid cavitating flows are performed around different two- and three-dimensional geometries, namely the NACA0012, NACA66(MOD) and two-element NACA4412-4415 hydrofoils, the hemispherical head shape body and the twisted NACA0009 hydrofoil, and the results are obtained over these geometries with the three cavitation models used for different flow conditions. The effects of different numerical parameters on the accuracy of the solution are also examined by a sensitivity study. The present results are compared with those of performed by other researchers which exhibit good agreement. It is indicated that the solution method adopted based on the preconditioned finite volume multiphase Euler flow solver on unstructured meshes is capable of accurately predicting the surface pressure distribution and the cavity shape over the arbitrary two- and three-dimensional geometries.

A nonlinear scheme preserving maximum principle for heterogeneous anisotropic diffusion equation

We present a new nonlinear method in order to preserve the maximum principle for the diffusion problem with heterogeneous anisotropic coefficient. It is well-known that for diffusion problems with general discontinuous coefficients, the existing cell-centered finite volume schemes on general meshes cannot unconditionally satisfy the discrete maximum principle (DMP), i.e., there are always certain severe restrictions on the mesh-cell geometry or locations of discontinuity in order to preserve DMP. To deal with the issue we propose a nonlinear method of modifying discrete flux of linear second-order schemes into a new conservative flux. Then the resulting scheme keeps conservation and accuracy, and has the structure preserving maximum principle without imposing those severe restrictions. Specifically, we will start with the diamond scheme whose flux is linear and has second-order accuracy, but has no the structure preserving maximum principle. Then the nonlinear correction method is applied to get a scheme satisfying the discrete maximum principle unconditionally. We give some theoretical analysis including the coercivity property and a prior estimation. Numerical results show that the accuracy of our new scheme is superior to the existing scheme preserving maximum principle, and in some cases the accuracy can even be improved by one order of magnitude.

Higher-order cell-centered finite volume scheme for the simulation of elastic-plastic flows in 3D

Higher-order cell-centered finite volume scheme for the simulation of elastic-plastic flows in 3D

An Interpolation-Free Cell-Centered Finite Volume Scheme for 3D Anisotropic Convection-Diffusion Equations on Arbitrary Polyhedral Meshes

An Interpolation-Free Cell-Centered Finite Volume Scheme for 3D Anisotropic Convection-Diffusion Equations on Arbitrary Polyhedral Meshes

An all Froude high order IMEX scheme for the shallow water equations on unstructured Voronoi meshes

We propose a novel numerical method for the solution of the shallow water equations in different regimes of the Froude number making use of general polygonal meshes. The fluxes of the governing equations are split such that advection and acoustic-gravity sub-systems are derived, hence separating slow and fast phenomena. This splitting allows the nonlinear convective fluxes to be discretized explicitly in time, while retaining an implicit time marching for the acoustic-gravity terms. Consequently, the novel schemes are particularly well suited in the low Froude limit of the model, since no numerical viscosity is added in the implicit solver. Besides, stability follows from a milder CFL condition which is based only on the advection speed and not on the celerity. High order time accuracy is achieved using the family of semi-implicit IMEX Runge-Kutta schemes, while high order in space is granted relying on two discretizations: (i) a cell-centered finite volume (FV) scheme for the nonlinear convective contribution on the polygonal cells; (ii) a staggered discontinuous Galerkin (DG) scheme for the solution of the linear system associated to the implicit discretization of the pressure sub-system. Therefore, three different meshes are used, namely a polygonal Voronoi mesh, a triangular subgrid and a staggered quadrilateral subgrid. The novel schemes are proved to be Asymptotic Preserving (AP), hence a consistent discretization of the limit model is retrieved for vanishing Froude numbers, which is given by the so-called “lake at rest” equations. Furthermore, the novel methods are well-balanced by construction, and this property is also demonstrated. Accuracy and robustness are then validated against a set of benchmark test cases with Froude numbers ranging in the interval Fr≈[10−6;5], hence showing that multiple time scales can be handled by the novel methods.

A cell-centered finite volume scheme for the diffusive–viscous wave equation on general polygonal meshes

Seismic wave equations based on numerical simulation have become effective tools in geological exploration. Considering the frequency dependence of reflections and fluid saturation in porous mediums, the diffusive–viscous wave theory is necessary to study. In this paper, a cell-centered finite volume scheme for the diffusive–viscous wave equation is proposed on general distorted polygonal meshes. Numerical experiments are provided to demonstrate the convergence rate of the errors in the discrete L2 norm and interpret the effectiveness by a simulation of the actual geological exploration point.

Effects of adiabatic and heat-conducting inserts in natural convection in inclined enclosures

Natural convection in a closed cavity with block-free, adiabatic and heat-conducting blocks was assessed for different inclination angles. The enclosure has vertical walls at different temperatures and adiabatic horizontal walls. The simulations were performed for the laminar regime and two-dimensional approximation. A cell-centered finite volume scheme was used to discretize the governing equations. This work addresses the influence of the Rayleigh number and cavity inclination in the flow pattern, temperature distribution, and local/global Nusselt numbers. It was observed that the largest heat transfer rates take place for heat-conducting blocks and inclination angles owing to higher velocity gradients.

A linearity-preserving technique for finite volume schemes of anisotropic diffusion problems on polygonal meshes

In this work we derive a cell-centered finite volume scheme by a linearity-preserving technique. It solves matrix equation obtained by linearity-preserving criterion and gets the desired approximations. The scheme introduces both cell-centered unknowns and vertex unknowns. The latter are auxiliary and eliminated by the surrounding cell-centered unknowns with a new vertex interpolation algorithm derived by the presented linearity-preserving technique. The proposed technique is flexible and is expected to design algorithms for 3D diffusion problems. Nearly optimal accuracy is found from the numerical experiments. More interesting is that the new vertex interpolation algorithm outperforms some commonly used linearity-preserving algorithms on Kershaw meshes and distorted meshes which have proved difficult for most algorithms.

A finite volume scheme preserving the invariant region property for the coupled system of FitzHugh-Nagumo equations on distorted meshes

In this paper, we propose and analyze a finite volume scheme preserving the invariant region property (IRP) for the coupled system of equations of FitzHugh-Nagumo type on distorted meshes. We employ a nonlinear cell-centered finite volume scheme to approximate the flux and implicit Euler scheme to approximate the time derivative. The IRP and existence of solutions are proved for the nonlinear finite volume scheme. We also propose a nonlinear iteration method which adds a specifically designed penalization term to ensure the IRP on each iteration step. Numerical experiments and the comparison with the nine-point finite volume scheme are presented to verify the IRP of our scheme.

A positivity-preserving and free energy dissipative hybrid scheme for the Poisson-Nernst-Planck equations on polygonal and polyhedral meshes

This paper develops a positivity-preserving and free energy dissipative hybrid scheme for the multi-dimensional Poisson-Nernst-Planck equations with multiple ions on unstructured polygonal and polyhedral meshes. It is built on the linear virtual element method for the Poisson-type equation for the electrostatic potential and a cell-centered finite volume scheme for the Nernst-Planck equations for the ion concentrations, and preserves the positivity of the ion concentrations, the mass conservation, the free energy dissipation and steady-state properties. The existence of the numerical solutions is also discussed. Numerical results demonstrate the effectiveness of the proposed two- and three-dimensional schemes.

A cell-centered implicit-explicit Lagrangian scheme for a unified model of nonlinear continuum mechanics on unstructured meshes

A cell-centered implicit-explicit updated Lagrangian finite volume scheme on unstructured grids is proposed for a unified first-order hyperbolic formulation of continuum fluid and solid mechanics, namely the Godunov-Peshkov-Romenski (GPR) model. The scheme provably respects the stiff relaxation limits of the continuous model at the fully discrete level, thus it is asymptotic preserving. Furthermore, the GCL is satisfied by a compatible discretization that makes use of a nodal solver to compute vertex-based fluxes that are used both for the motion of the computational mesh as well as for the time evolution of the governing PDEs. Second order of accuracy in space is achieved using a TVD piecewise linear reconstruction, while an implicit-explicit (IMEX) Runge-Kutta time discretization allows the scheme to obtain higher accuracy also in time. Particular care is devoted to the design of a stiff ODE solver, based on approximate analytical solutions of the governing equations, that plays a crucial role when the visco-plastic limit of the model is approached. We demonstrate the accuracy and robustness of the scheme on a wide spectrum of material responses covered by the unified continuum model that includes inviscid hydrodynamics, viscous heat conducting fluids, elastic and elasto-plastic solids in multidimensional settings.

Finite volume schemes of evolutionary diffusion equations based on harmonic averaging interpolation

This paper focuses on developing cell-centered finite volume schemes on arbitrary distorted and unstructured meshes for evolutionary diffusion equations. On each mesh cell, by using the cell center and three harmonic averaging points on cell boundaries as the interpolation points, we approximate the normal flux in four different ways. To make these finite volume schemes cell-centered ones, we eliminate the intermediate variables in the flux expression by the harmonic averaging interpolation procedure. We discretize the time derivative term by the backward Euler scheme. Under some proper assumptions on mesh, we prove the stability and error estimate of the scheme theoretically. Considering the continuous, discontinuous, and heterogeneous diffusion tensors, even for the nonlinear diffusion equations, we construct four numerical experiments to investigate the convergence and robustness. Several different unstructured triangular, quadrilateral, and polygonal meshes are adopted. The numerical results imply that the former two schemes are robust for any kind of mesh and any kind of diffusion tensor, and they can obtain second-order convergence in $L^2$ norm and first-order convergence in $H^1$ norm; but the latter two schemes are more dependent on mesh.

A 3D cell-centered ADER MOOD Finite Volume method for solving updated Lagrangian hyperelasticity on unstructured grids

In this paper, we present a conservative cell-centered Lagrangian Finite Volume scheme for solving the hyperelasticity equations on unstructured multidimensional grids. The starting point of the present approach is the cell-centered FV discretization named EUCCLHYD and introduced in the context of Lagrangian hydrodynamics. Here, it is combined with the a posteriori Multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure robustness and stability at shock waves with piecewise linear spatial reconstruction. The ADER (Arbitrary high order schemes using DERivatives) approach is adopted to obtain second-order of accuracy in time. This strategy has been successfully tested in a hydrodynamics context and the present work aims at extending it to the case of hyperelasticity. Here, the hyperelasticity equations are written in the updated Lagrangian framework and the dedicated Lagrangian numerical scheme is derived in terms of nodal solver, Geometrical Conservation Law (GCL) compliance, subcell forces and compatible discretization. The Lagrangian numerical method is implemented in 3D under MPI parallelization framework allowing to handle genuinely large meshes. A relatively large set of numerical test cases is presented to assess the ability of the method to achieve effective second order of accuracy on smooth flows, maintaining an essentially non-oscillatory behavior and general robustness across discontinuities and ensuring at least physical admissibility of the solution where appropriate. Pure elastic neo-Hookean and non-linear materials are considered for our benchmark test problems in 2D and 3D. These test cases feature material bending, impact, compression, non-linear deformation and further bouncing/detaching motions.

Upstream mobility finite volumes for the Richards equation in heterogenous domains

This paper is concerned with the Richards equation in a heterogeneous domain, each subdomain of which is homogeneous and represents a rocktype. Our first contribution is to rigorously prove convergence toward a weak solution of cell-centered finite-volume schemes with upstream mobility and without Kirchhoff’s transform. Our second contribution is to numerically demonstrate the relevance of locally refining the grid at the interface between subregions, where discontinuities occur, in order to preserve an acceptable accuracy for the results computed with the schemes under consideration.

The cell-centered positivity-preserving finite volume scheme for 3D anisotropic diffusion problems on distorted meshes

In this paper, a new cell-centered positivity-preserving finite volume scheme is proposed for the 3D anisotropic diffusion problems on distorted meshes. The primary unknowns and auxiliary unknowns are used in this scheme. First, the one-sided flux on each cell facet is discretized by a fixed stencil. Then, the nonlinear two-point flux approximation is applied to get the cell-centered discrete scheme. Besides, a second-order interpolation algorithm is constructed to eliminate the auxiliary unknowns in flux expressions. Furthermore, an improved Anderson acceleration method is given to accelerate the convergence of Picard iterations. Finally, several numerical examples are presented to show the accuracy and efficiency of the proposed scheme.

Positivity preserving and entropy consistent approximate Riemann solvers dedicated to the high-order MOOD-based Finite Volume discretization of Lagrangian and Eulerian gas dynamics

In this paper we propose to revisit the notion of simple Riemann solver both in Lagrangian and Eulerian coordinates following the seminal work of Gallice ”Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates” in Numer. Math., 94, 2003. We provide in this work the relation between the Eulerian and Lagrangian forms of systems of conservation laws in 1D. Then an approximate (simple) Lagrangian Riemann solver for gas dynamics is derived based on the notions of positivity preservation and entropy control. Its Eulerian counterpart is further deduced. Next we build the associated 1D first-order accurate cell-centered Lagrangian Godunov-type Finite Volume scheme and show numerically its behaviors on classical test cases. Then using the Lagrangian–Eulerian relationships, we derive and test the Eulerian Godunov-type Finite Volume scheme, which inherits by construction the properties of the Lagrangian solver in terms of positivity preservation and well-defined CFL condition. At last we extend this Eulerian scheme to arbitrary orders of accuracy using a Runge–Kutta time discretization, polynomial reconstruction and an a posteriori MOOD limiting strategy. Numerical tests are carried out to assess the robustness, accuracy, and essentially non-oscillatory properties of the numerical methods.

Об использовании элементно-центрированной конечно-объёмной схемы на призматических сетках в пристеночном слое

We consider the cell-centered finite-volume scheme with the quasi-one-dimensional reconstruction and generalize it to anisotropic prismatic meshes suitable for high-Reynolds-number problems. We offer a new algorithm of flux computation based on the reconstruction along the wall surface, whereas in the original schemes it was along the tangent to the wall surface. We also study how does the curvature of mesh elements influence the accuracy if taken into account.