Rosenbrock Methods for Partial Differential Equations and Fractional Orders of Convergence

Abstract

The authors apply Rosenbrock methods for the solution of initial value problems for linear partial differential equations $u_t (x,t) = \mathcal {L}(x,\partial )u(x,t) + f(x,t)$. Under weak assumptions on the operator $\mathcal {L}$ and the source function f, a sharp lower bound for the order of convergence of these numerical methods is given. It is further shown that the order of convergence is, in general, fractional. It depends on the $L^r$-norm used to estimate the global error. This analysis also applies to systems arising from spatial discretization of partial differential equations by finite differences or finite element techniques. Numerical examples illustrate the results.