Convergence of Runge-Kutta Methods Applied to Linear Partial Differential-Algebraic Equations

摘要

We apply Runge-Kutta methods to linear partial differential-algebraicequations of the form $Au_t(t,x) + B(u_{xx}(t,x)+ru_x(t,x))+Cu(t,x) = f(t,x)$,where $A,B,C\in\R^{n,n}$ and the matrix $A$ is singular. We prove that undercertain conditions the temporal convergence order of the fully discrete schemedepends on the time index of the partial differential-algebraic equation. Inparticular, fractional orders of convergence in time are encountered.Furthermore we show that the fully discrete scheme suffers an order reductioncaused by the boundary conditions. Numerical examples confirm the theoreticalresults.