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

TL;DR

This paper investigates how Runge-Kutta methods perform when applied to linear partial differential-algebraic equations with singular matrices, revealing convergence order dependencies and reductions due to boundary conditions.

Contribution

It provides a theoretical analysis of convergence orders for Runge-Kutta methods on PDE-DAEs, including fractional orders and boundary-induced order reduction, supported by numerical validation.

Findings

Convergence order depends on the time index of the PDE-DAE.

Fractional convergence orders can occur in time.

Boundary conditions cause order reduction in the scheme.

Abstract

We apply Runge-Kutta methods to linear partial differential-algebraic equations 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 under certain conditions the temporal convergence order of the fully discrete scheme depends on the time index of the partial differential-algebraic equation. In particular, fractional orders of convergence in time are encountered. Furthermore we show that the fully discrete scheme suffers an order reduction caused by the boundary conditions. Numerical examples confirm the theoretical results.