A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme

We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.     

Finite difference reaction–diffusion equations with nonlinear diffusion coefficients

Summary. A monotone iterative method for numerical solutions of a class of finite difference reaction-diffusion equations with nonlinear diffusion coefficient is presented. It is shown that by using an upper solution or a lower solution as the initial iteration the corresponding sequence converges monotonically to a unique solution of the finite difference system. It is also shown that the solution of the finite difference system converges to the solution of the continuous equation as the mesh size decreases to zero. Received February 18, 1998 / Revised version received April 21, 1999 / Published online February 17, 2000     

An error estimate for the finite difference approximation to degenerate convection–diffusion equations

We consider semi-discrete first-order finite difference schemes for a nonlinear degenerate convection?Cdiffusion equations in one space dimension, and prove an L ¹ error estimate. Precisely, we show that the ${L^1_{\rm{loc}}}$ difference between the approximate solution and the unique entropy solution converges at a rate ${\mathcal{O}(\Delta x^{1/11})}$ , where ${\Delta x}$ is the spatial mesh size. If the diffusion is linear, we get the convergence rate ${\mathcal{O}(\Delta x^{1/2})}$ , the point being that the ${\mathcal{O}}$ is independent of the size of the diffusion.     

Analysis of a BDF–DGFE scheme for nonlinear convection–diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection–diffusion equation. We employ a combination of the discontinuous Galerkin finite element method for the space semi-discretization and the k-step backward difference formula for the time discretization. The diffusive and stabilization terms are treated implicitly whereas the nonlinear convective term is treated by a higher order explicit extrapolation method, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the discrete L ^∞(L ²)-norm and the L ²(H ¹)-seminorm with respect to the mesh size h and time step τ for k = 2,3. Numerical examples verifying the theoretical results are presented. This work is a part of the research project MSM 0021620839 financed by the Ministry of Education of the Czech Republic and was partly supported by the Grant No. 316/2006/B-MAT/MFF of the Grant Agency of the Charles University Prague. The research of M. Vlasák was supported by the project LC06052 of the Ministry of Education of the Czech Republic (Jindřich Nečas Center for Mathematical Modelling).     

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product $f(\mathbf{A})\mathbf{b}$, where $\mathbf{A}$ is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.     

Standard finite difference scheme for a singularly perturbed elliptic convection–diffusion equation on a rectangle under computer perturbations

A singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε (ε ∈ (0, 1]) is considered on a rectangle. As applied to this equation, a standard finite difference scheme on a uniform grid is studied under computer perturbations. This scheme is not ε-uniformly stable with respect to perturbations. The conditions imposed on a “computing system” are established under which a converging standard scheme (referred to as a computer difference scheme) remains stable.     

Convergence of a nonlinear finite difference scheme for the Kuramoto–Tsuzuki equation

A nonlinear finite difference scheme is studied for solving the Kuramoto–Tsuzuki equation. Because the maximum estimate of the numerical solution can not be obtained directly, it is difficult to prove the stability and convergence of the scheme. In this paper, we introduce the Brouwer-type fixed point theorem and induction argument to prove the unique existence and convergence of the nonlinear scheme. An iterative algorithm is proposed for solving the nonlinear scheme, and its convergence is proved. Based on the iterative algorithm, some linearized schemes are presented. Numerical examples are carried out to verify the correction of the theory analysis. The extrapolation technique is applied to improve the accuracy of the schemes, and some interesting results are obtained.     

Accelerated monotone scheme for finite difference equations concerning steady-state prey–predator interactions

In previous work, by adapting a suitable finite difference method to a particular monotone scheme, the authors and A. Lazer have studied the numerical solution of a system of semilinear elliptic partial differential equations which determines the equilibria of the Volterra–Lotka equations describing prey–predator interactions with diffusion. In this paper, in order to improve the efficiency of the method, we show how Newton's method can be successfully combined with the previous scheme to greatly accelerate the convergence. In some particularly ‘difficult’ problems, the new method reduces the average number of iterations necessary to generate each element of the monotone sequences from 15 to about 3.     

On the stability of explicit finite difference methods for advection–diffusion equations

In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi-discretization of the ADE leads to a conditionally stable fully discretized method as long as the time-integrator is at least first-order accurate, whereas high-order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples.     

Robust convergence of a compact fourth-order finite difference scheme for reaction–diffusion problems

We consider a singularly perturbed one-dimensional reaction–diffusion problem with strong layers. The problem is discretized using a compact fourth order finite difference scheme. Altough the discretization is not inverse monotone we are able to establish its maximum-norm stability and to prove its pointwise convergence on a Shishkin mesh. The convergence is uniform with respect to the perturbation parameter. Numerical experiments complement our theoretical results.     

Traveling wave solutions of advection–diffusion equations with nonlinear diffusion

We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c∈]_c?,+∞[$c\in ]c_{?},+\infty [$, where _c?>0$c_{?}>0$ is explicitly computed but may not be optimal. We also prove that a free boundary hypersurface separates a region where u=0$u=0$ and a region where u>0$u>0$, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region u>0$u>0$, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed.     

Superconvergence of finite element method for singularly perturbed convection–diffusion equations in 1D

In this article, a singularly perturbed convection–diffusion equation is solved by a linear finite element method on a Shishkin mesh. By means of an analysis exploiting symmetries in the convective term of the bilinear form, a new superconvergence rate, which improves the existing result, is obtained.     

Stability and convergence of difference scheme for nonlinear evolutionary type equations

A finite difference scheme is derived for the initial-boundary problem for the nonlinear equation system $$\frac{\partial u}{\partial t}=A\frac{\partial^{2}u}{\partial x^{2}}+f(u),$$ where A is a complex diagonal matrix, f is a complex vector function. The stability and convergence in discrete L ^∞-norm of proposed Crank-Nicolson type finite difference schemes is proved. No restrictions on the ratio of time and space grid steps are assumed. Some numerical experiments have been conducted in order to validate the theoretical results.