## 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.

### MSC:

 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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