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Open Boundary Conditions for Nonlinear Initial Boundary Value Problems

Jan Nordström

TL;DR

This paper develops energy-stable open boundary conditions for nonlinear IBVPs by framing the boundary contribution through a diagonalized, nonlinear surface term and deriving a two-parameter boundary form $BU-G = S^{-1}(A^- - R A^+)U - G = 0$ that can be weakly enforced via a lifting operator. It shows how to reduce parameter complexity from $R,S,\\Sigma$ to $R,S$, and extends the theory to include viscous terms, enabling energy bounds using only nonlinear surface data. The method delivers a simpler and more practically implementable boundary treatment applicable to shallow-water, Euler, and Navier–Stokes systems, with provable energy stability and broader applicability.

Abstract

We present a straightforward energy stable weak implementation procedure of open boundary conditions for nonlinear initial boundary value problems. It simplifies previous work and its practical implementation.

Open Boundary Conditions for Nonlinear Initial Boundary Value Problems

TL;DR

This paper develops energy-stable open boundary conditions for nonlinear IBVPs by framing the boundary contribution through a diagonalized, nonlinear surface term and deriving a two-parameter boundary form that can be weakly enforced via a lifting operator. It shows how to reduce parameter complexity from to , and extends the theory to include viscous terms, enabling energy bounds using only nonlinear surface data. The method delivers a simpler and more practically implementable boundary treatment applicable to shallow-water, Euler, and Navier–Stokes systems, with provable energy stability and broader applicability.

Abstract

We present a straightforward energy stable weak implementation procedure of open boundary conditions for nonlinear initial boundary value problems. It simplifies previous work and its practical implementation.

Paper Structure

This paper contains 7 sections, 2 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Short summary of previous results
  3. The energy rates
  4. The nonlinear boundary conditions
  5. Previous and updated new general formulation of nonlinear boundary conditions
  6. Extension to include viscous terms
  7. Summary

Key Result

Proposition 2.2

The IBVP (eq:nonlin) with $C+C^T = 0$ has an energy rate that only depends on (eq:nonlin_BC).

Theorems & Definitions (10)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Lemma 3.2
  • Remark 3.3
  • proof