Finite Difference Approximation of Free Discontinuity Problems

We approximate functionals depending on the gradient of $u$ and on the behaviour of $u$ near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, $\Gamma$-convergence, and a compactness result which implies, in particular, the convergence of minima and minimizers.Comment: 39 pages. to appear on Proc. Royal Soc. Edinb. Ser.

Comparison of truncation error of finite-difference and finite-volume formulations of convection terms

Judging by errors in the computational-fluid-dynamics literature in recent years, it is not generally well understood that (above first-order) there are significant differences in spatial truncation error between formulations of convection involving a finite-difference approximation of the first derivative, on the one hand, and a finite-volume model of flux differences across a control-volume cell, on the other. The difference between the two formulations involves a second-order truncation-error term (proportional to the third-derivative of the convected variable). Hence, for example, a third (or higher) order finite-difference approximation for the first-derivative convection term is only second-order accurate when written in conservative control-volume form as a finite-volume formulation, and vice versa

Holistic projection of initial conditions onto a finite difference approximation

Modern dynamical systems theory has previously had little to say about finite difference and finite element approximations of partial differential equations (Archilla, 1998). However, recently I have shown one way that centre manifold theory may be used to create and support the spatial discretisation of \pde{}s such as Burgers' equation (Roberts, 1998a) and the Kuramoto-Sivashinsky equation (MacKenzie, 2000). In this paper the geometric view of a centre manifold is used to provide correct initial conditions for numerical discretisations (Roberts, 1997). The derived projection of initial conditions follows from the physical processes expressed in the PDEs and so is appropriately conservative. This rational approach increases the accuracy of forecasts made with finite difference models.Comment: 8 pages, LaTe

Numerical Solution of the Two-Phase Obstacle Problem by Finite Difference Method

In this paper we consider the numerical approximation of the two-phase membrane (obstacle) problem by finite difference method. First, we introduce the notion of viscosity solution for the problem and construct certain discrete nonlinear approximation system. The existence and uniqueness of the solution of the discrete nonlinear system is proved. Based on that scheme, we propose projected Gauss-Seidel algorithm and prove its convergence. At the end of the paper we present some numerical simulations.Comment: Free Boundary Problem, Two-Phase Membrane Problem, Two-Phase Obstacle Problem, Finite Difference Metho

Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

We obtain non-symmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton Jacobi Bellman Equations by introducing a new notion of consistency. We apply our general results to various schemes including finite difference schemes, splitting methods and the classical approximation by piecewise constant controls