Runge-Kutta Approximation of Quasi-Linear Parabolic Equations

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Abstract
We study the convergence properties of implicit Runge-Kutta methods applied to time discretization of parabolic equations with time- or solution-dependent 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 Runge-Kutta method. The second one is based on estimates for linear time-invariant equations and uses Fourier and perturbation techniques. It applies to $A(\theta )$-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.