Diagonally Implicit Runge-Kutta Methods for Nonlinear Dissipative Systems under Convex Functional in Vector Space

Source abstract

This paper is concerned with the contractivity and convergence of a class of diagonally implicit Runge-Kutta (DIRK) methods for nonlinear evolution equations governed by $\omega$-dissipative vector fields in a vector space and seminormed space, respectively. The exponential contractivity and long time convergence of this class of methods are studied by employing their error growth functions. B2-stable DIRK methods are showed to preserve unconditionally the exponential contractivity of $\omega$-dissipative vector fields under a nonnegative convex functional which can be norm, semi-norm, convex energy functional, non-negativity functional, et al. The unconditional contractivity-preserving property of high order DIRK methods is also proved for strongly $\omega$-dissipative vector fields in general vector space. The convergence results of these DIRK methods on infinite integration intervals are therefore obtained for these problems in seminormed space. Some applications of the main results to several DIRK methods are supplied. Numerical tests for the perturbed porous medium vector field and $\alpha$-Laplacian are provided to illustrate the contractivity property and accuracy of the proposed methods.