Nonlinear Finite Volume Scheme Research Articles

An arbitrary Lagrangian–Eulerian positivity-preserving finite volume scheme for radiation hydrodynamics equations in the equilibrium-diffusion limit

In this paper, an arbitrary Lagrangian–Eulerian (ALE) positivity-preserving finite volume scheme is constructed for radiation hydrodynamics equations (RHE) in the equilibrium-diffusion limit. Firstly, the integral form equations of RHE in the ALE framework are presented. Then, the operator-splitting method is applied to divide the equations into hyperbolic part and parabolic part. In addition, a second-order positivity-preserving finite volume scheme is constructed for the hyperbolic part based on MUSCL reconstruction. The vertex velocity is obtained by the predictor–corrector strategy. The Winslow method is applied to improve the quality of the Lagrangian mesh. Furthermore, a nonlinear positivity-preserving finite volume scheme suitable for distorted mesh is proposed for the parabolic part. Finally, some numerical examples are given to show the accuracy and reliability of the numerical scheme.

A symmetric and coercive finite volume scheme preserving the discrete maximum principle for anisotropic diffusion equations on star-shaped polygonal meshes

A nonlinear vertex-centered finite volume scheme that preserves the discrete maximum principle (DMP) is proposed for anisotropic diffusion equations on polygonal meshes. The new scheme is constructed from a vertex-centered linearity-preserving scheme combined with a nonlinear correction technique. The key ingredient is the introduction of the correction coefficient and the limiter. Particularly, the choice of the limiter is discussed and two new and effective limiters are proposed. Different from most existing nonlinear DMP-preserving schemes, the new nonlinear scheme does not rely on the nonlinear combination of linear flux approximations and requires no interpolation of the auxiliary unknowns. More important is that, it can be proved theoretically that the new scheme is symmetric and coercive on general polygonal meshes with star-shaped cells. Moreover, other theoretical results such as the DMP property and stability of the correction scheme are also presented. Numerical experiments demonstrate the accuracy, efficiency, and DMP property of the new scheme on general polygonal meshes with heterogeneous and anisotropic diffusion tensors.

On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations

The aim of this work is to develop a conservative, positivity-preserving (PP), nonlinear finite volume (FV) scheme for the multi-term nonlocal Nagumo-type equations on distorted meshes. These equations involve anisotropic diffusion tensor coefficients, and the solution exhibits a weak singularity at t=0. To achieve this, we combine the L1 scheme on a graded mesh with a PP-FV scheme on randomly distorted meshes, resulting in a conservative, positivity-preserving nonlinear L1-PP-FV scheme. By employing a nonlinear-splitting technique, we derive a series of simple nonlinear discrete equations that can be solved implicitly. We also provide proof that both the proposed FV scheme and the nonlinear iteration (NI) scheme are capable of maintaining positivity. Finally, we present numerical results to validate our theoretical analysis.

Error estimate of the cell-centered nonlinear positivity-preserving two-point flux approximation schemes

We present the error estimate of nonlinear positivity-preserving finite volume schemes for diffusion problems. These schemes are also named by nonlinear two-point flux approximation (NTPFA) methods. Due to their conservation and monotonicity, such schemes have attracted wide attention. Under suitable regularity assumptions on the exact solution, we prove a discrete H1 error estimate through the analysis of consistent errors. Some numerical examples are presented to demonstrate the theoretical results.

Compatible and energy conserving multi-material arbitrary Lagrangian Eulerian scheme for multi-group radiation hydrodynamics simulations

A compatible, bounds preserving and energy conserving multi-material arbitrary Lagrangian Eulerian (ALE) scheme suitable for an unstructured mesh is developed for multi-group radiation hydrodynamics (RHD) simulation of various high energy density physics (HEDP) phenomena involving large material deformations, such as intense thermal radiation or high-power laser driven systems. In order to study very short time scale HEDP phenomena, the electron and ion energy equations (with the effect of electron-ion energy relaxation) are solved separately with an assumption of single macroscopic fluid velocity for each species (two-temperature hydrodynamics model). The governing multi-material hydrodynamics equations are solved by using a higher-order cell-centered hydrodynamics (CCH) scheme. In multi-material flows, the remapping step introduces mixed computational cells in which multiple materials exist in a single cell. Therefore, material interfaces in mixed cells are required to be reconstructed and tracked. In our implementation, this is achieved by using moment-of-fluid (MOF) method. The information of these interfaces is further used to formulate an interface aware sub-scale dynamics (IASSD) scheme for multi-material closure in mixed computational cells. The IASSD scheme is generalized for the two-temperature hydrodynamics model in which separate equations are used to update the electron and ion energies. The multi-group radiation transport on an unstructured mesh is computed using a cell-centered, monotonic, nonlinear finite volume (NLFV) scheme. The flux-limited diffusion (FLD) approximation is used to recover the free-streaming limit of the radiation propagation in optically thin regions. In the ALE step, the electron, ion and radiation energies due to MG-RHD model are conservatively remapped using a center-of-mass (CM) reconstruction of the fields and a compatible and energy conserving remapping scheme. An exact integration based on the intersection between the new (rezoned) and old meshes is used in the remap step. Similar procedure is followed for the remap of cell mass and cell-centered momentum components. In this paper, the details of a compatible, bounds preserving and conservative ALE scheme developed for MG-RHD are described. The remapping of relevant physical quantities due to multi-physics models implemented in the code such as non-local electron energy transport are also discussed. Several validation test problems including laser driven shock propagation in multi-material geometries are provided to demonstrate the accuracy and performance of the ALE scheme implemented.

A positivity-preserving finite volume scheme for multi-group neutron diffusion kinetics equations with delayed neutrons

Neutron diffusion kinetics equation contains a diffusion type equation and a set of ordinary differential equations, which form a coupled system. In this paper, we give a positivity theorem for neutron diffusion kinetics equation with delayed neutrons and propose a nonlinear positivity-preserving finite volume scheme on general star-shaped quadrilateral meshes, which can deal with discontinuous and anisotropy diffusion coefficient problems. The existence of solution for the nonlinear system is proved under certain constraint on time step size, which is independent of spatial mesh size. Three nonlinear iteration strategies are introduced to solve the nonlinear system. Numerical experiments are performed to demonstrate that the scheme is positivity-preserving and keeps approximately second order accuracy in space. The computational costs of three nonlinear iteration strategies are compared. Another example is also present to show some constraint on time step size is necessary for the convergence of the nonlinear iteration.

Numerical analysis of a nonlinear discrete duality finite volume scheme for Leray-Lions type elliptic problems in Orlicz spaces

In this work, we develop a finite volume approximation for general nonlinear Leray-Lions problems in the Orlicz-Sobolev framework. We prove the existence and uniqueness and some a priori estimate of the approximate solution. We establish a discrete version of Poincaré inequality and a result of discrete compactness which allows us to prove the convergence towards the weak solution of the continuous problem. Some numerical tests are provided on general meshes.

Weighted positive nonlinear finite volume method for dominated anisotropic diffusive equations

In this paper, we propose a new positive finite volume scheme for degenerate parabolic equations with strongly anisotropic diffusion tensors. The key idea is to approximate the fluxes thanks to a weighted-centered scheme depending on the sign of the stiffness coefficients. More specifically, we employ a centered discretization for the mobility-like function when the transmissibilities are positive and a weighted harmonic scheme in regions where the negative transmissibilities occur. This technique prevents the formation of undershoots which entails the positivity-preserving of the approach. Also, the scheme construction retains the main elements, namely the coercivity and compactness estimates, allowing the existence and the convergence of the nonlinear finite volume scheme under general assumptions on the physical inputs, the nonlinearities, and the mesh. The numerical implementation of the methodology shows that the lower bound on the discrete solution is respected and optimal convergence rates are recovered on strongly anisotropic various test-cases.

A Time-Continuous Embedding Method for Scalar Hyperbolic Conservation Laws on Manifolds

A time-continuous (tc-)embedding method is first proposed for solving nonlinear scalar hyperbolic conservation laws with discontinuous solutions (shocks and rarefaction waves) on codimension 1, connected, smooth, and closed manifolds (surface PDEs or SPDEs in {mathbb {R}}^2 and {mathbb {R}}^3). The new embedding method improves upon the classical closest point (cp-)embedding method, which requires re-establishments of the constant-along-normal (CAN-)property of the extension function at every time step, in terms of accuracy and efficiency, by incorporating the CAN-property analytically and explicitly in the embedding equation. The tc-embedding SPDEs are solved by the second-order nonlinear central finite volume scheme with a nonlinear minmod slope limiter in space, and the third-order total variation diminished Runge-Kutta scheme in time. An adaptive nonlinear essentially non-oscillatory polynomial interpolation is used to obtain the solution values at the ghost cells. Numerical results in solving the linear wave equation and the Burgers’ equation show that the proposed tc-embedding method has better accuracy, improved resolution, and reduced CPU times than the classical cp-embedding method. The Burgers’ equation, the traffic flow problem, and the Buckley-Leverett equation are solved to demonstrate the robust performance of the tc-embedding method in resolving fine-scale structures efficiently even in the presence of a shock and the essentially non-oscillatory capturing of shocks and rarefaction waves on simple and complex shaped one-dimensional manifolds. Burgers’ equation is also solved on the two-dimensional torus-shaped and spherical-shaped manifolds.

A cost-effective nonlinear extremum-preserving finite volume scheme for highly anisotropic diffusion on Cartesian grids, with application to radiation belt dynamics

We construct a new nonlinear finite volume (FV) scheme for highly anisotropic diffusion equations, that satisfies the discrete minimum-maximum principle. The construction relies on the linearized scheme satisfying less restrictive monotonicity conditions than those of an M-matrix, based on a weakly regular matrix splitting and using the Cartesian structure of the mesh (extension to quadrilateral meshes is also possible). The resulting scheme, obtained by expressing fluxes as nonlinear combinations of linear fluxes, has a larger stencil than other nonlinear positivity preserving or minimum-maximum principle preserving schemes. Its larger “linearized” stencil, closer to the actual complete stencil (that includes unknowns appearing in the convex combination coefficients), enables a faster convergence of the Picard iterations used to compute the solution of the scheme. Steady state dimensionless numerical tests as well as simulations of the highly anisotropic diffusion in electron radiation belts show a second order of convergence of the new scheme and confirm its computational efficiency compared to usual nonlinear FV schemes.

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.

Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes

We propose a positivity-preserving finite volume scheme on non-conforming quadrilateral distorted meshes with hanging nodes for subdiffusion equations, where the differential equations have a sum of time-fractional derivatives of different orders, and the typical solutions of the problem have a weak singularity at the initial time \(t=0\) for given smooth data. In this paper, a positivity-preserving nonlinear method with centered unknowns is obtained by the two-point flux technique, where a new method to handling vertex unknown including hanging nodes is the highlight of our paper. For each time derivative, we apply the L1 scheme on a temporal graded mesh. Especially, the existence of a solution is strictly proved for the nonlinear system by applying the Brouwer’s fixed point theorem. Numerical results show that the proposed positivity-preserving method is effective for strongly anisotropic and heterogeneous full tensor subdiffusion coefficient problems.

The VEM-based positivity-preserving conservative scheme for radiation diffusion problems on generalized polyhedral meshes

In this paper, for the 3D radiation diffusion equation on generalized polyhedral meshes, we construct a positivity-preserving conservative scheme based on the virtual element method (VEM). Firstly, the equation is discretized by finite volume, and the one-sided flux on the cell-face is discretized by using the fixed stencil of all vertices. Then, the nonlinear two-point flux approximation is used to obtain the equations about cell-centered unknowns and vertex unknowns; the lowest order VEM solves the vertex values of the mesh. Finally, the conservative equations containing only cell-centered unknowns are solved. This scheme ensures the high accuracy of VEM and embodies the local conservation and positivity-preserving. Compared with the existing nonlinear positivity-preserving finite volume schemes, the new scheme does not need nonlinear iteration for linear problems. For nonlinear problems, it only needs to use the VEM to solve the nonlinear system. Numerical results show that the new scheme can achieve the optimal convergence order and maintain the nonnegativity of the solution for generalized polyhedral meshes and arbitrary diffusion tensors.

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.

Analysis of Accurate and Stable Nonlinear Finite Volume Scheme for Anisotropic Diffusion Equations with Drift on Simplicial Meshes

This work addresses the development and analysis of a second-order accurate finite volume scheme for parabolic equations with anisotropy on general simplicial meshes. The discretization involves only vertex unknowns without processing additional ones. The scheme construction makes use of a nonlinear transformation of the linear elliptic term. Two propositions are mainly presented for the approximation of the mobility function at the interfaces. The existence of positive solutions for the discrete system is guaranteed thanks to the proved a priori estimates. The energy dissipation of the scheme is moreover ensured. The convergence of the approach is established. Numerical tests are given to show the efficiency, accuracy and robustness of the proposed approach, with respect to the anisotropy, while a particular emphasis is set on the effects of the approximate mobility. They also confirm the obtained theoretical results, especially the decay of the free energy when time grows.

Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes

Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes

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.

Nonlinear finite volume discretization for transient diffusion problems on general meshes

A nonlinear discrete duality finite volume scheme is proposed for time-dependent diffusion equations. The model example is written in a new formulation giving rise to similar nonlinearities for both the diffusion and the potential functions. A natural finite volume discretization is built on this particular problem's structure. The fluxes are generically approximated thanks to a key fractional average. The point of this strategy is to promote coercivity and scheme's stability simultaneously. The existence of positive solutions is guaranteed. The theoretical convergence of the nonlinear scheme is established using practical compactness tools. Numerical results are performed in order to highlight the second order accuracy of the methodology and the positiveness of solutions on distorted meshes.

Convergence of nonlinear finite volume schemes for two-phase porous media flow on general meshes

Abstract In this work we present an abstract finite volume discretization framework for incompressible immiscible two-phase flow through porous media. A priori error estimates are derived that allow us to prove the existence of discrete solutions and to establish the proof of convergence for schemes belonging to this framework. In contrast to existing publications the proof is not restricted to a specific scheme and it assumes neither symmetry nor linearity of the flux approximations. Two nonlinear schemes, namely a nonlinear two-point flux approximation and a nonlinear multipoint flux approximation, are presented, and some properties of these schemes, e.g. saturation bounds, are proven. Furthermore, the numerical behavior of these schemes (e.g. accuracy, coercivity, efficiency or saturation bounds) is investigated for different test cases for which the coercivity is checked numerically.