Nonlinear and Linearised Primal and Dual Initial Boundary Value Problems: When are they Bounded? How are they Connected?
Jan Nordström
TL;DR
This work addresses the mismatch between nonlinear energy conservation/bounds and their linearised counterparts in initial boundary value problems. It introduces a skew-symmetric primal formulation and a non-standard linearisation that together preserve energy properties for both nonlinear and linearised problems, including their duals, and shows how appropriate boundary conditions ensure energy bounds. The theory is illustrated through Burgers' equation, 2D incompressible Euler, 3D cylindrical Euler, and 2D shallow water equations, highlighting boundary-treatment nuances and the impact of variable transformations. Finally, the authors connect the continuous theory to SBP-SAT numerical schemes, demonstrating automatic energy stability and conservation for both linear and nonlinear primal and dual problems when formulated in SBP form.
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.