Runge - Kutta Approximation of Quasi - Linear Parabolic Equations

We study the convergence properties of implicit Runge-Kutta methods applied to time discretization of parabolic equations with timeor solutiondependent operator. Error bounds are derived in the energy norm. The convergence analysis uses two different approaches. The first, technically simpler approach relies on energy estimates and requires algebraic stability of the RungeKutta method. The second one is based on estimates for linear time-invariant equations and uses Fourier and perturbation techniques. It applies to A(9)stable Runge-Kutta methods and yields the precise temporal order of convergence. This order is noninteger in general and depends on the type of boundary conditions. Introduction In this paper we investigate the approximation properties of implicit RungeKutta methods applied to time discretization of parabolic equations with timeor solution-dependent operator. Apart from some results in Crouzeix's thesis [3], this appears not to have been studied previously. There are, however, a number of papers dealing with the backward Euler or Crank-Nicolson method, and a few papers studying multistep methods. These papers fall into two groups, depending on whether the results are obtained from1 (A) estimates for linear time-invariant equations coupled with perturbation techniques [3, 18, 21, 26], or (B) energy estimates, e.g. [7, 27] (cf. also [14]). Both approaches turn out to be useful also in the context of Runge-Kutta methods, and to offer different merits. They work with different assumptions about the equation (A: resolvent bounds, B: Gârding's inequality) and require different stability conditions on the part of the methods (A: ^(ö)-stability, B: i?-stability or algebraic stability). When they apply, energy estimates provide far simpler stability and convergence proofs. It seems, however, that they do not yield the noninteger temporal convergence order which is actually observed in computations and can be explained via approach (A). When it comes to modified Runge-Kutta methods, in particular linearly implicit methods [24], there Received by the editor October 7, 1993 and, in revised form, April 8, 1994. 1991 Mathematics Subject Classification. Primary 65M12, 65M15, 65M20, 65J10, 65J15.