Continuous and Discontinuous Galerkin Time Stepping Methods for Nonlinear Initial Value Problems with Application to Finite Time Blow-Up

TL;DR

This paper develops continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems, proving existence and uniqueness of solutions, and applying the theory to finite time blow-up problems with a new time step algorithm.

Contribution

It introduces new techniques for proving existence of solutions independent of polynomial degree and develops a time step algorithm for blow-up time computation.

Findings

Existence of solutions is independent of approximation order.

A time step selection algorithm for blow-up problems is proposed.

Convergence of the blow-up time computation is established.

Abstract

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to nonlinear initial value problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.