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The Onsager principle and structure preserving numerical schemes

Huangxin Chen, Hailiang Liu, Xianmin Xu

TL;DR

This work leverages the Onsager principle to demonstrate its efficacy in formulating partial differential equation models for diverse gradient flow systems and provides a robust basis for developing numerical schemes that uphold crucial physical properties.

Abstract

We present a natural framework for constructing energy-stable time discretization schemes. By leveraging the Onsager principle, we demonstrate its efficacy in formulating partial differential equation models for diverse gradient flow systems. Furthermore, this principle provides a robust basis for developing numerical schemes that uphold crucial physical properties. Within this framework, several widely used schemes emerge naturally, showing its versatility and applicability.

The Onsager principle and structure preserving numerical schemes

TL;DR

This work leverages the Onsager principle to demonstrate its efficacy in formulating partial differential equation models for diverse gradient flow systems and provides a robust basis for developing numerical schemes that uphold crucial physical properties.

Abstract

We present a natural framework for constructing energy-stable time discretization schemes. By leveraging the Onsager principle, we demonstrate its efficacy in formulating partial differential equation models for diverse gradient flow systems. Furthermore, this principle provides a robust basis for developing numerical schemes that uphold crucial physical properties. Within this framework, several widely used schemes emerge naturally, showing its versatility and applicability.
Paper Structure (14 sections, 1 theorem, 125 equations, 13 figures)

This paper contains 14 sections, 1 theorem, 125 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Onsager principle as a modeling tool
  3. Systems for nonconserved parameters
  4. Systems for conserved parameters
  5. Application Examples
  6. Time-discretization based on the Onsager principle
  7. Discretization for systems with nonconserved parameters
  8. Discretization for systems with conserved parameters
  9. Time discretization by examples
  10. Numerical solutions
  11. The discretized conserved problem
  12. Solution by optimization algorithms
  13. Simulations
  14. Conclusion

Key Result

Proposition 1

Suppose $(\boldsymbol{u}^{k + 1}, \boldsymbol{m}^{k + 1})$ is a minimizer of the minimizing problem e:discOnsCsv2, we then have and $\boldsymbol{u}^{k + 1}$satisfies the mass conservation equation that

Figures (13)

  • Figure 1: The solutions of the PNP equation. First Row: the distribution of $u_1$ at various time $t=0.0005,0.0025,0.005$; Second Row: the distribution of $u_2$ at various time $t=0.0005,0.0025,0.005$; Last Row: the potential $\psi$ at various time $t=0.0005,0.0025,0.005$.
  • Figure 2: The change of the total energy with respect to time.
  • Figure 3: The change of the total mass of the two components with respect to time.
  • Figure 4: Initial distributions of wetting-phase saturation and permeability in Example 2.
  • Figure 5: Energy dissipation with time in Example 2.
  • ...and 8 more figures

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Claim 1
  • Proposition 1
  • Remark 3
  • Remark 4
  • Remark 5