Monotone Finite Volume Research Articles

A monotone diffusion scheme for 3D general meshes: Application to radiation hydrodynamics in the equilibrium diffusion limit

We propose in this article a monotone finite volume diffusion scheme on 3D general meshes for the radiation hydrodynamics. Primary unknowns are averaged value over the cells of the mesh. It requires the evaluation of intermediate unknowns located at the vertices of the mesh. These vertex unknowns are computed using an interpolation method. In a second step, the scheme is made monotone by combining the computed fluxes. It allows to recover monotonicity, while making the scheme nonlinear. This scheme is inserted into a radiation hydrodynamics solver and assessed on radiation shock solutions on deformed meshes.

On the accuracy of the finite volume approximations to nonlocal conservation laws

AbstractIn this article, we discuss the error analysis for a certain class of monotone finite volume schemes approximating nonlocal scalar conservation laws, modeling traffic flow and crowd dynamics, without any additional assumptions on monotonicity or linearity of the kernel $$\mu $$ μ or the flux f. We first prove a novel Kuznetsov-type lemma for this class of PDEs and thereby show that the finite volume approximations converge to the entropy solution at the rate of $$\sqrt{\Delta t}$$ Δ t in $$L^1(\mathbb {R})$$ L 1 ( R ) . To the best of our knowledge, this is the first proof of any type of convergence rate for this class of conservation laws. We also present numerical experiments to illustrate this result.

A monotone finite volume element scheme for diffusion equations on arbitrary polygonal grids

We develop a monotone finite volume element scheme for the diffusion problem on arbitrary polygonal grids inspired by the scheme in the literature ([35]). The main contributions of this paper include three aspects. Firstly, some auxiliary lines or points are introduced for any polygonal element so that it can be partitioned into two or more triangles, and one novel type of its control volume is concomitantly designed. In this paper, we mainly discuss quadrilateral grids. Secondly, some piecewise linear finite element spaces are imported for the trial spaces. Fictitious elements, continuous extensions and the nonlinear two-point flux idea can all be inherited from the triangular case. Therefore, the monotonicity of the scheme is easily proved only by the assembling of some triangles' element stiff matrices. Thirdly, the agreeable adaptability for distorted grids is completely borned. The strict convexity restriction on the grids can also be removed, the concave or severely distorted polygonal (such as quadrilateral) grids are transformed as the convex and general regular triangles. Finally, numerical results confirm its monotonicity, accuracy and stability.

A monotone finite volume scheme for single phase flow with reactive transport in anisotropic porous media

This paper deals with development and analysis of a finite volume scheme for a nonlinear parabolic equation modeling a single phase flow with reactive solute transport, including equilibrium sorption, in heterogeneous porous media. We present a second-order, finite volume scheme by combining a nonlinear two point flux approximation for the diffusion term, and an upwind method for the advective term on unstructured meshes. It is shown that, under a CFL condition, this scheme ensures the non-negativity of the discrete solution, taking into account the anisotropy and heterogeneity of the domain. The existence of a solution for the discrete nonlinear system is studied, and the convergence of a fixed point method is proved. Our theoretical results are illustrated by numerical simulations in a bidimensional domain, in particular we show the second-order convergence rate for the numerical solution and confirm the monotonicity of the scheme.

Weakly monotone finite volume scheme for parabolic equations in strongly anisotropic media

Weakly monotone finite volume scheme for parabolic equations in strongly anisotropic media

A novel monotone finite volume element scheme for diffusion equations

We present a novel monotone finite volume element scheme for the diffusion problem on triangular grids. Firstly, one fictitious triangular element is introduced for each edge of any element, and another vertex of this fictitious element is on the outward extended straight line passing through two points: The vertex opposite to this edge and the barycenter of the element. The diffusive tensor and the solution in this element are continuously extended to its corresponding fictitious element. The fictitious element and the continuous extension not only avoid the phenomenon that the control volume segment passes through other elements, but also are helpful to the construction and theoretical analysis for our monotone scheme. Secondly, one new nonlinear two-point flux formulation is obtained for some part control volume edge whose another endpoint is the barycenter of the fictitious element, and this straight segment crosses through the common edge of the triangular element and its fictitious one. The approximation of κ∇u on this straight control volume edge is constructed by some weighted average with the gradients of two finite element functions, and the weighted coefficients are determined by the nonlinear two-point flux idea. Furthermore, we proved that the new scheme is monotone under some conditions: the diffusion tensor and exact solution functions are properly smooth, and the partition step size h is properly small, and the solutions on three nodes of any element are not nonnegative, and furthermore not simultaneously zero, and the nodes of the fictitious element are properly chosen. We also proved that the corresponding coefficient matrix is of M-matrix. Finally, we carry on some typical experiments. Numerical results verify the monotonicity of this new scheme, and confirm its accuracy and stability.

A robust, interpolation‐free and monotone finite volume scheme for diffusion equations on arbitrary quadrilateral meshes

AbstractA novel monotone finite volume (FV) scheme on arbitrary quadrilateral meshes, such as extremely distorted meshes with concave, twisted, or even negative volume cells, is proposed to solve diffusion equations with discontinuous tensor coefficients. The first advantage of our scheme is that no auxiliary unknown is needed thus the efficiency and accuracy are improved. The second advantage is that the convex decomposition of the conormal vector relies only on the locations of cell centers and reduces geometric restrictions on cell shapes to preserve positivity. The difficulties of constructing discrete normal fluxes across the material interface are overcome by introducing primary unknowns on the opposite side of the interface into the discrete stencil with some virtual adjustments on their positions, based on the continuity of normal flux and tangent gradients. Numerical tests demonstrate that our new scheme is second‐order accurate, unconditionally monotone, and comparisons with some existing methods in terms of efficiency and accuracy are also given.

Parallel domain decomposition schemes based on finite volume element discretization for nonsteady-state diffusion equations on distorted meshes

In this paper, several domain decomposition algorithms are considered to solve parallelly non-steady diffusion equations. The contributions of this work are threefold. The first one is that we present effective parallel domain decomposition schemes underlying the classical and monotone finite volume element discretization. These schemes adopt a three-stage technique, which is also used in the difference method and the finite volume method. Positivity of the parallel scheme based on monotone finite volume element method is proved. The second one is that we propose a new domain decomposition computational strategy to improve the correction step in prediction-correction frame such that the whole parallel process is simplified. The third one is that we consider a parallel algorithm build on quadratic finite volume element method, which preserves local conservation and has higher convergence rates. These parallel schemes are of intrinsic parallelism since we apply the prediction-correction technique to the interface of subdomains coming from an artificial division of computational domain or multi-medium physical domain. The proposed parallel schemes need only local communication among neighboring processors. Numerical results are presented to illuminate the accuracy, stability and parallelism of the parallel schemes.

A monotone finite volume element scheme for diffusion equations on triangular grids

We present a monotone finite volume element scheme for the diffusion problem on triangular grids, which stems from some idea about the two-point flux in the literature [15]. A new nonlinear two-point flux formulation is obtained for some part of one control volume edge, and this straight edge crosses through some common edge of two neighboring triangular elements where two inner points from these neighboring elements are chosen and linked so that it does not cross through other triangle elements even if the meshes are severely distorted. The approximation of κ∇u in some quadrilateral region including this straight control volume edge, are obtained by some weighted average with the gradient of some finite element functions, and the weighted coefficients are determined by the nonlinear two-point flux idea. Furthermore, we prove that the new scheme is monotone under some conditions that one modified barycenter dual partition Ωh⁎ is chosen for some given partition Ωh and the exact solution u∈C1(Ω), and the corresponding coefficient matrix is of M-matrix. Finally, we carry on some typical experiments. Numerical results verify the monotonicity of this new scheme, and also confirm the accuracy and stability.

Monotonicity correction for second order element finite volume methods of anisotropic diffusion problems

We apply the monotonicity correction to second-order element finite volume methods, which include quadratic element on triangular meshes and biquadratic element on quadrilateral meshes, for anisotropic diffusion problems, and obtain a second-order monotone finite volume scheme. The main ideas of monotonicity correction are as follows: When we formulate the second-order element finite volume schemes, we need to calculate line integrals on the boundary line segment of the dual element, and regard these line integrals as numerical fluxes, which are in the form of multi-point flux. Among these numerical fluxes, we can separate a two-point flux structure on both sides of the line segment. Therefore, we can apply nonlinear monotone correction to these numerical fluxes, then we obtain a corrected second order element finite volume schemes whose stiffness matrix is monotone matrix, so the first high order unconditional positivity-preserving finite volume schemes are obtained by us. Moreover, we analyze the truncation error of the corrected numerical fluxes. Furthermore, for the time dependent problem, we also apply monotone correction to time derivative term. Numerical experiments show that the corrected second order element finite volume schemes have monotonicity, and maintain the convergence order of the original schemes, which are second order in H1 norm and third order in L2 norm.

A finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes

We construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes, where the strict convexity restriction on the meshes is removed. The main contributions of this paper include three aspects. Firstly, the classical finite volume element (C-FVE) method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation. In fact, the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme. Secondly, a new monotonicity correction is suggested, based on which we obtain a monotone finite volume element (M-FVE) method. The resulting M-FVE method still keeps the local conservation and is easy for implementation. Finally, we analyze theoretically the truncation error and the monotonicity for this scheme. Besides, the existence of a solution to this nonlinear scheme is proved by applying the Brouwer's fixed point theorem. Numerical results demonstrate that the M-FVE method has the approximate second-order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes.

Convergence Rates of Monotone Schemes for Conservation Laws with Discontinuous Flux

We prove that a class of monotone finite volume schemes for scalar conservation laws with discontinuous flux converge at a rate of $\sqrt{\Delta x}$ in $\mathrm{L}^1$, whenever the flux is strictly monotone in $u$ and the spatial dependency of the flux is piecewise constant with finitely many discontinuities. We also present numerical experiments to illustrate the main result. To the best of our knowledge, this is the first proof of any type of convergence rate for numerical methods for conservation laws with discontinuous, nonlinear flux. Our proof relies on convergence rates for conservation laws with initial and boundary value data. Since those are not readily available in the literature we establish convergence rates in that case en passant in the Appendix.

A New Interpolation for Auxiliary Unknowns of the Monotone Finite Volume Scheme for 3D Diffusion Equations

A New Interpolation for Auxiliary Unknowns of the Monotone Finite Volume Scheme for 3D Diffusion Equations

The Optimal Convergence Rate of Monotone Schemes for Conservation Laws in the Wasserstein Distance

In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone finite volume approximations of conservation laws. For compactly supported, $\Lip^+$-bounded initial data they showed a first-order convergence rate in the Wasserstein distance. Our main result is to prove that this rate is optimal. We further provide numerical evidence indicating that the rate in the case of $\Lip^+$-unbounded initial data is worse than first-order.

A Monotone Finite Volume Scheme with Fixed Stencils for 3D Heat Conduction Equation

A Monotone Finite Volume Scheme with Fixed Stencils for 3D Heat Conduction Equation

Calibration of a sedimentation model through a continuous genetic algorithm

ABSTRACTIn this contribution we consider the problem of flux identification in a scalar conservation law modelling the phenomenon of sedimentation. The experimental observation data used for the calibration consist of a solid concentration profile at a fixed time. The identification problem is formulated as an optimization one, where the distance between the profiles of the model simulation and observation data is minimized by a least squares cost function. The direct problem is approximated by a monotone finite volume scheme. The numerical solution of the calibration problem is obtained by a continuous genetic algorithm. Numerical results are presented in order to validate the efficiency of the proposed algorithm.

A monotone finite volume method for time fractional Fokker-Planck equations

We develop a monotone finite volume method for the time fractional Fokker-Planck equations and theoretically prove its unconditional stability. We show that the convergence rate of this method is order 1 in space and if the space grid becomes sufficiently fine, the convergence rate can be improved to order 2. Numerical results are given to support our theoretical findings. One characteristic of our method is that it has monotone property such that it keeps the nonnegativity of some physical variables such as density, concentration, etc.

Monotone Finite Volume Schemes for Diffusion Equation with Imperfect Interface on Distorted Meshes

In this paper, we prove the solution of diffusion equation with imperfect interface is positivity-preserving and a monotone finite volume method is presented to obtain the nonnegative solution on distorted mesh. Motivated by Sheng and Yuan (J Comput Phys 231:3739–3754, 2012), the discrete normal flux on interface is defined by using an extended stencil and introducing two auxiliary points to distinguish the discontinuities of the unknowns on both sides of the interface. The resulting finite volume scheme is locally conservative and has only cell-centered unknowns. Moreover, it is proved to be monotone. The numerical results show that the method obtains second order convergent rate in \(L_2\) norms for solutions on quadrilateral and triangular meshes.

A Monotone Finite Volume Scheme with Second Order Accuracy for Convection-Diffusion Equations on Deformed Meshes.

A Monotone Finite Volume Scheme with Second Order Accuracy for Convection-Diffusion Equations on Deformed Meshes.

A finite volume scheme with improved well modeling in subsurface flow simulation

We present the latest enhancement of the nonlinear monotone finite volume method for the near-well regions. The original nonlinear method is applicable for diffusion, advection-diffusion, and multiphase flow model equations with full anisotropic discontinuous permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which reduces to the conventional two-point flux approximation (TPFA) on cubic meshes but has much better accuracy for the general case of non-orthogonal grids and anisotropic media. The latest modification of the nonlinear method takes into account the nonlinear (e.g., logarithmic) singularity of the pressure in the near-well region and introduces a correction to improve accuracy of the pressure and the flux calculation. In this paper, we consider a linear version of the nonlinear method waiving its monotonicity for sake of better accuracy. The new method is generalized for anisotropic media, polyhedral grids and nontrivial cases such as slanted, partially perforated wells or wells shifted from the cell center. Numerical experiments show noticeable reduction of numerical errors compared to the original monotone nonlinear FV scheme with the conventional Peaceman well model or with the given analytical well rate.