Positivity-preserving Finite Volume Scheme Research Articles

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.

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.

An efficient semi-implicit friction source term treatment for modeling overland flow

In this study, a new numerical treatment of the friction source term is proposed for modeling shallow water flows over complex bed topography. We design a novel semi-implicit temporal discretization technique based on Taylor series expansion of friction source terms. A well-balanced positivity-preserving finite volume scheme on unstructured triangular grids is used to solve the shallow water model. We present a series of numerical tests to demonstrate the robustness and effectiveness of the proposed numerical model where we used rapidly varying flows over complex topography as well as flows involving small values of water depth and wet/dry zones. The simulation results of the numerical model based on the proposed approach for friction source terms are compared to experimental data and to those obtained by using other existing techniques. The proposed numerical model is stable and second-order accurate and our predictions are in good agreement with experimental observations.

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.

A positivity-preserving RD-FV scheme for diffusion problems on triangular meshes

In this paper, we propose a positivity-preserving residual distribution based finite volume scheme (RD-FV) for steady diffusion problems on unstructured triangular meshes. In this scheme, we use the RD method to obtain the auxiliary values of vertex, then adopt the positivity-preserving finite volume scheme to update the cell-centered values for diffusion problems, in which a nonlinear two-point flux approximation is utilized to obtain nonnegative solution. Thus a positivity-preserving conservative property for the RD-FV scheme is guaranteed. Several numerical experiments show the optimal convergence rates for the numerical solution, that is, approximate second-order accuracy for the solution and first-order accuracy for the flux, and demonstrate the positivity-preserving property of our new 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.

A Positivity-Preserving Finite Volume Scheme for Nonequilibrium Radiation Diffusion Equations on Distorted Meshes.

In this paper, we propose a new positivity-preserving finite volume scheme with fixed stencils for the nonequilibrium radiation diffusion equations on distorted meshes. This scheme is used to simulate the equations on meshes with both the cell-centered and cell-vertex unknowns. The cell-centered unknowns are the primary unknowns, and the element vertex unknowns are taken as the auxiliary unknowns, which can be calculated by interpolation algorithm. With the nonlinear two-point flux approximation, the interpolation algorithm is not required to be positivity-preserving. Besides, the scheme has a fixed stencil and is locally conservative. The Anderson acceleration is used for the Picard method to solve the nonlinear systems efficiently. Several numerical results are also given to illustrate the efficiency and strong positivity-preserving quality of the scheme.

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.

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.

A positivity-preserving finite volume scheme for convection–diffusion equation on general meshes

A new positivity-preserving finite volume scheme is proposed for solving the convection–diffusion equation on 2D or 3D distorted meshes. The nonlinear two-point flux approximation is applied to the discretization of diffusion flux. The discretization of convection flux is based on the second-order upwind method with a slope limiter. The cell centres are employed to define the primary unknowns. The cell vertexes are devoted to define the auxiliary unknowns, which can be computed from the primary unknowns. And the interpolation method of this scheme is not required to be positivity-preserving. Numerical results illustrate that the scheme is effective in solving the convection–diffusion problem and has second-order convergence rate on the distorted meshes.

Nonlinear Finite Volume Scheme Preserving Positivity for 2D Convection-Diffusion Equations on Polygonal Meshes

In this paper, a nonlinear finite volume scheme preserving positivity for solving 2D steady convection-diffusion equation on arbitrary convex polygonal meshes is proposed. First, the nonlinear positivity-preserving finite volume scheme is developed. Then, in order to avoid the computed solution beyond the upper bound, the cell-centered unknowns and auxiliary unknowns on the cell-edge are corrected. We prove that the present scheme can avoid the numerical solution beyond the upper bound. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results show that our scheme preserves the above conclusion and has second-order accuracy for solution.

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

A novel cell-centered finite volume scheme with positivity-preserving property for the anisotropic diffusion problems on general polyhedral meshes

In this paper, we propose a novel cell-centered positivity-preserving finite volume scheme for the anisotropic diffusion problems. The discretization of diffusion flux is based on the standard nonlinear two-point flux approximation. The cell vertexes are employed to define auxiliary unknowns. A positivity-preserving vertex interpolation algorithm is constructed to get the value of auxiliary unknowns. This interpolation algorithm has almost second order convergence rate on the distorted meshes. Numerical results illustrate that our scheme is efficient and practical in solving the 3D anisotropic diffusion problems.

A positivity-preserving finite volume scheme for three-temperature radiation diffusion equations

In this paper, a positivity-preserving finite volume scheme is proposed for the two-dimensional three-temperature (2-D 3-T) radiation diffusion equations. The cell centers are given to define the primary unknowns. The cell vertexes are used to define auxiliary unknowns, which can be computed by the primary unknowns. The nonlinear two-point flux approximation is applied to the discretization of diffusion flux. This scheme is not necessary the interpolation method with positivity-preserving property. Besides, this scheme is local conservative and second-order accurate on the distorted meshes. The existence of discrete solution for this scheme is proved. The stability analysis is also obtained with some assumptions. Numerical experiments indicate that the finite volume scheme is robust and well positive in solving the 2-D 3-T radiation diffusion equations.

Positivity-preserving finite volume scheme for compressible two-phase flows in anisotropic porous media: The densities are depending on the physical pressures

We are concerned with the approximation of solutions to a compressible two-phase flow model in porous media thanks to an enhanced control volume finite element discretization. The originality of the methodology consists in treating the case where the densities are depending on their own pressures without any major restriction neither on the permeability tensor nor on the mesh. Contrary to the ideas of [23], the point of the current scheme relies on a phase-by-phase “sub”-unpwinding approach so that we can recover the coercivity-like property. It allows on a second place for the preservation of the physical bounds on the discrete saturation. The convergence of the numerical scheme is therefore performed using classical compactness arguments. Numerical experiments are presented to exhibit the efficiency and illustrate the qualitative behavior of the implemented method.

Monotone nonlinear finite-volume method for challenging grids

This article presents a new positivity-preserving finite-volume scheme with a nonlinear two-point flux approximation, which uses optimization techniques for the face stencil calculation. The gradient is reconstructed using harmonic averaging points with the constraint that the sum of the coefficients included in the face stencils must be positive. We compare the proposed scheme to a nonlinear two-point scheme available in literature and a few linear schemes. Using two test cases, taken from the FVCA6 benchmarks, the accuracy of the scheme is investigated. Furthermore, it is shown that the scheme is linearity-preserving on highly complex corner-point grids. Moreover, a two-phase flow problem on the Norne formation, a geological formation in the Norwegian Sea, is simulated. It is demonstrated that the proposed scheme is consistent in contrast to the linear Two-Point Flux Approximation scheme, which is industry standard for simulating subsurface flow on corner-point grids.

A Positivity-Preserving Finite Volume Scheme for Heat Conduction Equation on Generalized Polyhedral Meshes

A Positivity-Preserving Finite Volume Scheme for Heat Conduction Equation on Generalized Polyhedral Meshes

A vertex-centered and positivity-preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids

We suggest a new positivity-preserving finite volume scheme for anisotropic diffusion problems on arbitrary polygonal grids. The scheme has vertex-centered, edge-midpoint and cell-centered unknowns. The vertex-centered unknowns are primary and have finite volume equations associated with them. The edge-midpoint and cell-centered unknowns are treated as auxiliary ones and are interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. Unlike most existing positivity-preserving schemes, the construction of the scheme is based on a special nonlinear two-point flux approximation that has a fixed stencil and does not require the convex decomposition of the co-normal. In order to solve efficiently the nonlinear systems resulting from the nonlinear scheme, Picard method and its Anderson acceleration are discussed. Numerical experiments demonstrate the second-order accuracy and well positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids. The high efficiency of the Anderson acceleration is also shown on reduction of the number of nonlinear iterations. Moreover, the proposed scheme does not have the so-called numerical heat-barrier issue suffered by most existing cell-centered and hybrid schemes. However, further improvements have to be made if the solution is very close to the machine precision and the mesh distortion is very severe.

A Second-Order Positivity-Preserving Finite Volume Scheme for Diffusion Equations on General Meshes

We propose a new positivity-preserving finite volume scheme for the anisotropic diffusion problems on general polygonal meshes based on a new nonlinear two-point flux approximation. The scheme uses both cell-centered and cell-vertex unknowns. The cell-vertex unknowns are treated as auxiliary ones and are computed by two second-order interpolation algorithms. Due to the new nonlinear two-point flux formulation, it is not required to replace the interpolation algorithm with positivity-preserving but usually low-order accurate ones whenever negative interpolation weights occur and it is also unnecessary to require the decomposition of the conormal vector to be a convex one. Moreover, the new nonlinear two-point flux approximation has a fixed stencil. These features make our scheme more flexible, easy for implementation, and different from other existing nonlinear schemes based on Le Potier's two-point flux approximation. The positivity-preserving property of our scheme is proved theoretically and numerical r...

Variable-order finite elements and positivity preservation for hyperbolic PDEs

A family of positivity-preserving finite element methods are considered for the solution of the advection equation in one space dimension. The approach uses B-spline spatial discretization methods in conjunction with forward Euler timestepping and a mass matrix iteration that preserves positivity. The method is compared against a positivity-preserving finite volume scheme on travelling wave and pulse examples including an inviscid Burgers' equation example.