Nonlinear and Linearised Primal and Dual Initial Boundary Value Problems: When are they Bounded? How are they Connected?

TL;DR

This paper explores the relationship between nonlinear and linearised initial boundary value problems, showing how specific formulations can ensure energy conservation and bounds are preserved across both versions and their duals, with implications for boundary conditions and numerical stability.

Contribution

It introduces a specific skew-symmetric formulation that maintains energy bounds and conservation in both nonlinear and linearised problems, clarifying their connection and impact on boundary conditions.

Findings

A skew-symmetric form leads to energy conservation in nonlinear problems.

A modified linearisation preserves energy bounds and conservation.

Formulations on summation-by-parts basis enable energy stable numerical schemes.

Abstract

Linearisation is often used as a first step in the analysis of nonlinear initial boundary value problems. The linearisation procedure frequently results in a confusing contradiction where the nonlinear problem conserves energy and has an energy bound but the linearised version does not (or vice versa). In this paper we attempt to resolve that contradiction and relate nonlinear energy conserving and bounded initial boundary value problems to their linearised versions and the related dual problems. We start by showing that a specific skew-symmetric form of the primal nonlinear problem leads to energy conservation and a bound. Next, we show that this specific form together with a non-standard linearisation procedure preserves these properties for the new slightly modified linearised problem. We proceed to show that the corresponding linear and nonlinear dual (or self-adjoint) problems also have bounds and conserve energy due to this specific formulation. Next, the implication of the new formulation on the choice of boundary conditions is discussed. A straightforward nonlinear and linear analysis may lead to a different number and type of boundary conditions required for an energy bound. We show that the new formulation shed some light on this contradiction. We conclude by illustrating that the new continuous formulation automatically lead to energy stable and energy conserving numerical approximations for both linear and nonlinear primal and dual problems if the approximations are formulated on summation-by-parts form.