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Nonlinear and linearised primal and dual initial boundary value problems: When are they bounded? How are they connected?

Nordström, Jan

Journal of computational physics, 2022-04, Vol.455, p.111001, Article 111001 [Periódico revisado por pares]

Cambridge: Elsevier Inc

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  • Título:
    Nonlinear and linearised primal and dual initial boundary value problems: When are they bounded? How are they connected?
  • Autor: Nordström, Jan
  • Assuntos: Approximation ; Boundary conditions ; Boundary value problems ; Dual problems ; Energy ; Energy conservation ; Energy stability ; Linear analysis ; Linearisation procedure ; Linearization ; Nonlinear analysis ; Nonlinear initial boundary value problems ; Skew-symmetric formulation ; Summation-by-parts
  • É parte de: Journal of computational physics, 2022-04, Vol.455, p.111001, Article 111001
  • Descrição: •We relate nonlinear energy bounded initial boundary value problems to their linearised versions.•We show that a specific skew-symmetric form of a nonlinear and linearised problem leads to energy conservation and a bound.•The skew-symmetric primal formulation preserves energy conservation and boundedness for the linear and nonlinear dual problems.•The implication of the new formulation on the choice of boundary conditions is discussed.•The skew-symmetric continuous formulation automatically leads to energy stable and energy conserving numerical approximations. 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 sheds some light on this contradiction. We conclude by illustrating that the new continuous formulation automatically leads 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.
  • Editor: Cambridge: Elsevier Inc
  • Idioma: Inglês
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  • Realização: ABCD-USP USP   Logos de Redes Sociais: Freepik

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