Convergence Rates for Semi-Discrete Splitting Approximations for Degenerate Parabolic Equations with Source Terms

Abstract

We study a semi-discrete splitting method for computing approximate viscosity solutions of the initial value problem for a class of nonlinear degenerate parabolic equations with source terms. It is fairly standard to prove that the semi-discrete splitting approximations converge to the desired viscosity solution as the splitting step Δt tends to zero. The purpose of this paper is, however, to consider the more difficult problem of providing a precise estimate of the convergence rate. Using viscosity solution techniques we establish the L∞ convergence rate $\mathcal{O}(\sqrt{\Delta t})$ for the approximate solutions, and this estimate is robust with respect to the regularity of the solutions. We also provide an extension of this result to weakly coupled systems of equations, and in the case of more regular solutions we recover the “classical” rate $\mathcal{O}(\Delta t)$ . Finally, we analyze in an example a fully discrete splitting method.