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Antisymmetry, pseudospectral methods, and conservative PDEs

Robert McLachlan, Nicolas Robidoux

Abstract

`Dual composition', a new method of constructing energy-preserving discretizations of conservative PDEs, is introduced. It extends the summation-by-parts approach to arbitrary differential operators and conserved quantities. Links to pseudospectral, Galerkin, antialiasing, and Hamiltonian methods are discussed.

Antisymmetry, pseudospectral methods, and conservative PDEs

Abstract

`Dual composition', a new method of constructing energy-preserving discretizations of conservative PDEs, is introduced. It extends the summation-by-parts approach to arbitrary differential operators and conserved quantities. Links to pseudospectral, Galerkin, antialiasing, and Hamiltonian methods are discussed.

Paper Structure

This paper contains 5 sections, 5 theorems, 16 equations.

Table of Contents

  1. Abstract.
  2. 1. Introduction
  3. 2. Discretizing conservative PDEs
  4. 3. Examples of the dual composition method
  5. 4. Quadrature of Hamiltonians

Key Result

Proposition 1

Let ${\mathfrak F}_0$ be natural for ${\mathcal{H}}$ and let $S$ be nonsingular. Then for every $u\in{\mathfrak F}_0$ there is a unique element $\overline{\frac{\delta {\mathcal{H}}}{\delta u}}\in{\mathfrak F}_1$ such that Its coordinate representation is $S^{-1}\nabla H$ where $H({\mathbf u}):={\mathcal{H}}(u_i f_i)$.

Theorems & Definitions (17)

  • Definition 1
  • Example 1
  • Example 2
  • Definition 2
  • Definition 3
  • Proposition 1
  • Proposition 2
  • Definition 4
  • Proposition 3
  • Proposition 4
  • ...and 7 more