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Nonlinear Boundary Conditions for Initial Boundary Value Problems with Applications in Computational Fluid Dynamics

Jan Nordström

TL;DR

This work develops a nonlinear boundary procedure that extends the classical linear characteristic boundary approach to nonlinear IBVPs with non-zero data, within a skew-symmetric framework. By combining energy and entropy analysis with SBP–SAT discretizations, it derives conditions under which boundary terms are bounded or dissipative, enabling energy stability for both hyperbolic and parabolic problems. The authors formulate a general boundary condition in transformed variables, introduce scaling by solution-dependent eigenvalues, and provide a systematic route (R,S,Σ) to ensure strong or weak nonlinear boundary enforcement. They extend the analysis to second-derivative terms and demonstrate provable stability for key CFD models: 2D incompressible Euler, shallow water equations, 2D compressible Euler, and incompressible Navier–Stokes. The framework supports both strong and weak imposition of non-zero boundary data and yields practical, provably stable procedures for high-fidelity CFD simulations with complex boundary inputs.

Abstract

We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems (IBVPs) with non-zero boundary data that lead to bounded solutions. The new boundary procedure is applied to nonlinear IBVPs in skew-symmetric form, including dissipative terms. The complete procedure has two main ingredients. In the first part (published in [1, 2]), the energy and entropy rate in terms of a surface integral with boundary terms was produced for problems with first derivatives. In this second part we complement it by adding second derivative terms and new nonlinear boundary procedures leading for boundary conditions with non-zero data. The new nonlinear boundary procedure generalise the well known characteristic boundary procedure for linear problems to the nonlinear setting. To introduce the procedure, a skew-symmetric scalar IBVP encompassing the linear advection equation and Burgers equation is analysed. Once the continuous analysis is done, we show that energy stable nonlinear discrete approximations follow by using summation-by-parts operators combined with weak boundary conditions. The scalar analysis is subsequently repeated for general nonlinear systems of equations. Finally, the new boundary procedure is applied to four important IBVPs in computational fluid dynamics: the incompressible Euler and Navier-Stokes, the shallow water and the compressible Euler equations.

Nonlinear Boundary Conditions for Initial Boundary Value Problems with Applications in Computational Fluid Dynamics

TL;DR

This work develops a nonlinear boundary procedure that extends the classical linear characteristic boundary approach to nonlinear IBVPs with non-zero data, within a skew-symmetric framework. By combining energy and entropy analysis with SBP–SAT discretizations, it derives conditions under which boundary terms are bounded or dissipative, enabling energy stability for both hyperbolic and parabolic problems. The authors formulate a general boundary condition in transformed variables, introduce scaling by solution-dependent eigenvalues, and provide a systematic route (R,S,Σ) to ensure strong or weak nonlinear boundary enforcement. They extend the analysis to second-derivative terms and demonstrate provable stability for key CFD models: 2D incompressible Euler, shallow water equations, 2D compressible Euler, and incompressible Navier–Stokes. The framework supports both strong and weak imposition of non-zero boundary data and yields practical, provably stable procedures for high-fidelity CFD simulations with complex boundary inputs.

Abstract

We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems (IBVPs) with non-zero boundary data that lead to bounded solutions. The new boundary procedure is applied to nonlinear IBVPs in skew-symmetric form, including dissipative terms. The complete procedure has two main ingredients. In the first part (published in [1, 2]), the energy and entropy rate in terms of a surface integral with boundary terms was produced for problems with first derivatives. In this second part we complement it by adding second derivative terms and new nonlinear boundary procedures leading for boundary conditions with non-zero data. The new nonlinear boundary procedure generalise the well known characteristic boundary procedure for linear problems to the nonlinear setting. To introduce the procedure, a skew-symmetric scalar IBVP encompassing the linear advection equation and Burgers equation is analysed. Once the continuous analysis is done, we show that energy stable nonlinear discrete approximations follow by using summation-by-parts operators combined with weak boundary conditions. The scalar analysis is subsequently repeated for general nonlinear systems of equations. Finally, the new boundary procedure is applied to four important IBVPs in computational fluid dynamics: the incompressible Euler and Navier-Stokes, the shallow water and the compressible Euler equations.
Paper Structure (22 sections, 5 theorems, 107 equations)

This paper contains 22 sections, 5 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Part 0: Illustration of the main concepts in a scalar setting: a roadmap for the paper
  3. Part 1: Producing energy and entropy rates
  4. Part 2: The nonlinear boundary conditions and weak implementation procedure
  5. Part 3: Semi-discrete nonlinear stability
  6. Part 1: Producing the energy and entropy rates
  7. Energy and entropy rates when $\epsilon \rightarrow 0$
  8. Energy and entropy rates when $\epsilon > 0$
  9. Part 2: The nonlinear boundary conditions and weak implementation procedure
  10. One significant difference between linear and nonlinear boundary conditions
  11. The general form of nonlinear boundary conditions for transformed variables
  12. Extension of the analysis to include second derivative terms
  13. Part 3: Semi-discrete nonlinear stability
  14. Application of the general theory to initial boundary value problems in CFD
  15. The 2D incompressible Euler equations
  16. ...and 7 more sections

Key Result

Proposition 3.2

The IBVP (eq:nonlin) with $\epsilon = 0$ for is energy conserving if It is energy bounded if it is energy conserving and the boundary conditions (eq:nonlin_BC) for $G=0$ lead to It is strongly energy bounded if it is energy conserving and the boundary conditions (eq:nonlin_BC) for $G \neq 0$ lead to where $G=G(\vec{x},t)$ is independent of the solution $U$.

Theorems & Definitions (31)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 3.1
  • Remark 3.1
  • Proposition 3.2
  • ...and 21 more