Runge-Kutta methods for partial differential equations and fractional orders of convergence

Abstract:

We apply Runge-Kutta methods to linear partial differential equations of the form ${u_t}(x,t) = \mathcal {L}(x,\partial )u(x,t) + f(x,t)$. Under appropriate assumptions on the eigenvalues of the operator $\mathcal {L}$ and the (generalized) Fourier coefficients of f, we give a sharp lower bound for the order of convergence of these methods. We further show that this order is, in general, fractional and that it depends on the ${L^r}$-norm used to estimate the global error. The 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.