Stiffly accurate Runge–Kutta methods for nonlinear evolution problems governed by a monotone operator

Abstract

Abstract. Stiffly accurate implicit Runge-Kutta methods are studied for the time discretisation of nonlinear first-order evolution equations. The equation is supposed to be governed by a time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fulfills a certain growth condition. It is proven that the piecewise constant as well as the piecewise linear interpolant of the time-discrete solution converges towards the exact weak solution, provided the Runge-Kutta method is consistent and satisfies a stability criterion that implies algebraic stability; examples are the Radau IIA and Lobatto IIIC methods. The convergence analysis is also extended to problems involving a strongly continuous perturbation of the monotone main part.

Time discretisation of monotone nonlinear evolution problems by the discontinuous Galerkin method

“…8.2] for the backward Euler method. For other time discretisation methods, we have recently been able to prove similar results (see [5,6] for the two-step backward differentiation formula (BDF) on an equidistant grid, [7] for the two-step BDF on a variable time grid, [8] for the ϑ -scheme on a variable time grid, and [9] for a class of stiffly accurate Runge-Kutta methods) although the assumptions on the underlying operator as well as the convergence results themselves differ from method to method. Moreover, in contrast to the techniques used there, the analysis of the discontinuous Galerkin method is indeed more involved and requires additional techniques due to their non-conforming character and the external approximation of the standard solution space for (1.1) by W I .…”

Unconditional convergence of DIRK schemes applied to dissipative evolution equations

“…Crandall and Liggett [6] have derived convergence for the implicit Euler method when (1) is merely defined on a Banach space. Other related results can be found in the paper by Rulla [17], who derives the optimal convergence order for the implicit Euler scheme when f is a subgradient, and the studies by Emmrich [8] and Emmrich and Thalhammer [9], where the weak convergence of the BDF2 scheme and of stiffly accurate Runge-Kutta methods is proven for monotone vector fields acting on Gelfand triples. Furthermore, for vector fields of the form f = f a + f b the convergence of various splitting schemes has been proven by Lions and Mercier [15] and by ourselves in [11].…”

On a Full Discretisation for Nonlinear Second-Order Evolution Equations with Monotone Damping: Construction, Convergence, and Error Estimates

“…In the context of nonlinear first-order in time evolution equations it is well known that certain stability properties of the numerical approximation are essential in order to be able to utilise the monotone and coercive structure of the underlying operator and to establish a convergence result. As a small excerpt of contributions for nonlinear first-order evolution equations governed by a monotone operator, we mention EMMRICH [14,15] providing a convergence analysis of the two-step backward differentiation formula and EMMRICH, THALHAMMER [18] introducing a stability criterium which enables to treat a subclass of stiffly accurate Runge-Kutta methods, see also the references given therein. Contrary, the construction of numerical methods for second-order evolution equations, especially for problems where no first-order damping term is present, primarily aims at the preservation of geometric structures such as symplecticity.…”