A numerical approach to degenerate parabolic equations

Summary. In this work we propose a numerical approach to solve some kind of degenerate parabolic equations. The underlying idea is based on the maximum principle. More precisely, we locally perturb the (initial and boundary) data instead of the nonlinear diffusion coefficients, so that the resulting problem is not degenerate. The efficiency of this method is shown analytically as well as numerically. The numerical experiments show that this new approach is comparable with the existing ones. Received January 20, 1999 / Revised version received February 28, 2000 / Published online July 25, 2001     

Finite element approximation of a fourth order nonlinear degenerate parabolic equation

Summary. We consider a fully practical finite element approximation of the fourth order nonlinear degenerate parabolic equation where generically for any given . An iterative scheme for solving the resulting nonlinear discrete system is analysed. In addition to showing well-posedness of our approximation, we prove convergence in one space dimension. Finally some numerical experiments are presented. Received July 29, 1997     

Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

Summary. One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation by a piecewise constant function using a discretization in space and time and a finite volume scheme. The convergence of to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on are used to prove the convergence, up to a subsequence, of to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of to{\it u}. Some on a model equation are shown. Received September 27, 2000 / Published online October 17, 2001     

Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Summary. Explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features: degeneracy, quasilinearity, full nonlinearity, and singularities. In particular, the equation of “motion by mean curvature” is included. The schemes are monotone and consistent, so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems. In addition, an intriguing new type of nonlocal problem is analyzed which is related to the schemes, and another very different sort of approximation is presented as well. Received January 10, 1995     

A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations

Summary. This paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the -norm in the situation, where the diffusion parameter is smaller or comparable to the mesh size. Numerical experiments underline the theoretical results. Received February 25, 1999 / Revised version received July 6, 1999 / Published online August 2, 2000     

Analysis of fully discrete approximations to parabolic problems with rough or distribution-valued initial data

Summary. We consider fully discrete approximations to a parabolic initial-boundary value problem with rough or distribution-valued initial data in two space dimensions. For discretization in time and space, we apply single step methods and the standard Galerkin method with piecewise linear test functions, respectively. For spatial discretization of the initial condition, we are however forced to use more involved constructions. Our main result is stability and error estimates of the discrete solutions. Received October 21, 1999 / Revised version received May 3, 2001 / Published online December 18, 2001     

Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations

Summary. We first analyse a semi-discrete operator splitting method for nonlinear, possibly strongly degenerate, convection-diffusion equations. Due to strong degeneracy, solutions can be discontinuous and are in general not uniquely determined by their data. Hence weak solutions satisfying an entropy condition are sought. We then propose and analyse a fully discrete splitting method which employs a front tracking scheme for the convection step and a finite difference scheme for the diffusion step. Numerical examples are presented which demonstrate that our method can be used to compute physically correct solutions to mixed hyperbolic-parabolic convection-diffusion equations. Received November 4, 1997 / Revised version received June 22, 1998     

An operator splitting method for nonlinear convection-diffusion equations

Summary. We present a semi-discrete method for constructing approximate solutions to the initial value problem for the -dimensional convection-diffusion equation . The method is based on the use of operator splitting to isolate the convection part and the diffusion part of the equation. In the case , dimensional splitting is used to reduce the -dimensional convection problem to a series of one-dimensional problems. We show that the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions. ReceivedFebruary 1, 1996 / Revised version received June 24, 1996     

Implicit-explicit multistep methods for quasilinear parabolic equations

Summary. Efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in [1]. In this note we present a refined analysis to allow more general nonlinearities. The abstract theory is applied to a quasilinear parabolic equation. Received March 10, 1997 / Revised version received March 2, 1998     

Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations

Summary. A general method for constructing high-order approximation schemes for Hamilton-Jacobi-Bellman equations is given. The method is based on a discrete version of the Dynamic Programming Principle. We prove a general convergence result for this class of approximation schemes also obtaining, under more restrictive assumptions, an estimate in of the order of convergence and of the local truncation error. The schemes can be applied, in particular, to the stationary linear first order equation in . We present several examples of schemes belonging to this class and with fast convergence to the solution. Received July 4, 1992 / Revised version received July 7, 1993     

Approximation with frequency filter for backward parabolic equations

Summary. We propose a numerical method for the initial (and boundary) value problem for the equation of the form where A is an unbounded, selfadjoint operator with negative spectrum. Roundoff errors in the numerical solution of such problem may generate a parasite term growing very quickly with time. To eliminate this parasite term, we apply a special finite difference equation with r free parameters. Similar ideas may be useful also for another numerically difficult differential problems. Received October 6, 1997 / revised version received November 26, 1998 / Published online October 16, 2000     

Convergence of Runge–Kutta approximations for parabolic problems with Neumann boundary conditions

Summary. We analyze a class of algebraically stable Runge–Kutta/standard Galerkin methods for inhomogeneous linear parabolic equations, with time–dependent coefficients, under Neumann boundary conditions, and derive an error bound of provided is bounded. Received June 25, 1994 / Revised version received February 26, 1996     

A priori error estimates for approximate solutions to convex conservation laws

Summary. We introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite–difference approximation calculated with the scheme of Engquist-Osher, Lax-Friedrichs, or Godunov. This technique is a discrete counterpart of the duality technique introduced by Tadmor [SIAM J. Numer. Anal. 1991]. The error is related to the consistency error of cell averages of the entropy weak solution. This consistency error can be estimated by exploiting a regularity structure of the entropy weak solution. One ends up with optimal error estimates. Received December 21, 2001 / Revised version received February 18, 2002 / Published online June 17, 2002     

Numerical validation of solutions of complementarity The nonlinear case

Summary. This paper proposes a validation method for solutions of nonlinear complementarity problems. The validation procedure performs a computational test. If the result of the test is positive, then it is guaranteed that a given multi-dimensional interval either includes a solution or excludes all solutions of the nonlinear complementarity problem. Received September 22, 2000 / Revised version received April 11, 2001 / Published online October 17, 2001     

On the stability of the unsmoothed Fourier method for hyperbolic equations

Summary. It has been a long open question whether the pseudospectral Fourier method without smoothing is stable for hyperbolic equations with variable coefficients that change signs. In this work we answer this question with a detailed stability analysis of prototype cases of the Fourier method. We show that due to weighted -stability, the -degree Fourier solution is algebraically stable in the sense that its amplification does not exceed . Yet, the Fourier method is weakly -unstable in the sense that it does experience such amplification. The exact mechanism of this weak instability is due the aliasing phenomenon, which is responsible for an amplification of the Fourier modes at the boundaries of the computed spectrum. Two practical conclusions emerge from our discussion. First, the Fourier method is required to have sufficiently many modes in order to resolve the underlying phenomenon. Otherwise, the lack of resolution will excite the weak instability which will propagate from the slowly decaying high modes to the lower ones. Second -- independent of whether smoothing was used or not, the small scale information contained in the highest modes of the Fourier solution will be destroyed by their amplification. Happily, with enough resolution nothing worse can happen. Received December 14, 1992/Revised version received March 1, 1993     

Convergence of fluctuation-splitting schemes for two dimensional scalar conservation laws with a kinetic solver

Summary. The “fluctuation-splitting schemes” (FSS in short) have been introduced by Roe and Sildikover to solve advection equations on rectangular grids and then extended to triangular grids by Roe, Deconinck, Struij... For a two dimensional nonlinear scalar conservation law, we consider the case of a triangular grid and of a kinetic approach to reduce the discretization of the nonlinear equation to a linear equation and apply a particular FSS called N-scheme. We show that the resulting scheme converges strongly in in a finite volume sense. Received February 25, 1997 / Revised version received November 8, 1999 / Published online August 24, 2000     

Convergence of a staggered Lax-Friedrichs scheme for nonlinear conservation laws on unstructured two-dimensional grids

Summary. Based on Nessyahu and Tadmor's nonoscillatory central difference schemes for one-dimensional hyperbolic conservation laws [16], for higher dimensions several finite volume extensions and numerical results on structured and unstructured grids have been presented. The experiments show the wide applicability of these multidimensional schemes. The theoretical arguments which support this are some maximum-principles and a convergence proof in the scalar linear case. A general proof of convergence, as obtained for the original one-dimensional NT-schemes, does not exist for any of the extensions to multidimensional nonlinear problems. For the finite volume extension on two-dimensional unstructured grids introduced by Arminjon and Viallon [3,4] we present a proof of convergence for the first order scheme in case of a nonlinear scalar hyperbolic conservation law. Received April 8, 2000 / Published online December 19, 2000