Runge-Kutta methods for partial differential equations and fractional orders of convergence. (English) Zbl 0769.65068

This elegant paper studies the (possibly fractional) order behaviour of implicit Runge-Kutta methods when applied to linear partial differential equations \(u_ t(x,t) = Lu(x,t) + f(x,t)\), \({u(0) = u_ 0}\), where \(L\) is a differential operator with a pure point spectrum of nonpositive real part.
An \(A\)-stable Runge-Kutta method with stage order \(q\) and order of consistency \(p\) is shown to have global order given by \(\alpha_ r = \min\{p,q + 2 + \nu_ r\}\), where \(\nu_ r\) depends on the \(L_ r\) norm used to estimate the global error. In the case of the one- dimensional heat equation \(u_ t = u_{xx} + a(x)g(t)\) on \(\Omega = (0,1)\) with \(a(x)\) differentiable and nonvanishing on \(\partial\Omega\), \(\nu_ r = {1\over 2r}\).
A similar analysis for nonhomogeneous boundary conditions is given – in which case \(q+2\) is replaced by \(q+1\) or (in the case of even stage Gauss methods) \(q\). The results are numerically verified for a simple semi- discretized problem.


65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65L05 Numerical methods for initial value problems involving ordinary differential equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
35K15 Initial value problems for second-order parabolic equations
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