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Two arbitrary-order constraint-preserving schemes for the Yang--Mills equations on polyhedral meshes

Jérôme Droniou, Jia Jia Qian

Abstract

Two numerical schemes are proposed and investigated for the Yang--Mills equations, which can be seen as a nonlinear generalisation of the Maxwell equations set on Lie algebra-valued functions, with similarities to certain formulations of General Relativity. Both schemes are built on the Discrete de Rham (DDR) method, and inherit from its main features: an arbitrary order of accuracy, and applicability to generic polyhedral meshes. They make use of the complex property of the DDR, together with a Lagrange-multiplier approach, to preserve, at the discrete level, a nonlinear constraint associated with the Yang--Mills equations. We also show that the schemes satisfy a discrete energy dissipation (the dissipation coming solely from the implicit time stepping). Issues around the practical implementations of the schemes are discussed; in particular, the assembly of the local contributions in a way that minimises the price we pay in dealing with nonlinear terms, in conjunction with the tensorisation coming from the Lie algebra. Numerical tests are provided using a manufactured solution, and show that both schemes display a convergence in $L^2$-norm of the potential and electrical fields in $\mathcal O(h^{k+1})$ (provided that the time step is of that order), where $k$ is the polynomial degree chosen for the DDR complex. We also numerically demonstrate the preservation of the constraint.

Two arbitrary-order constraint-preserving schemes for the Yang--Mills equations on polyhedral meshes

Abstract

Two numerical schemes are proposed and investigated for the Yang--Mills equations, which can be seen as a nonlinear generalisation of the Maxwell equations set on Lie algebra-valued functions, with similarities to certain formulations of General Relativity. Both schemes are built on the Discrete de Rham (DDR) method, and inherit from its main features: an arbitrary order of accuracy, and applicability to generic polyhedral meshes. They make use of the complex property of the DDR, together with a Lagrange-multiplier approach, to preserve, at the discrete level, a nonlinear constraint associated with the Yang--Mills equations. We also show that the schemes satisfy a discrete energy dissipation (the dissipation coming solely from the implicit time stepping). Issues around the practical implementations of the schemes are discussed; in particular, the assembly of the local contributions in a way that minimises the price we pay in dealing with nonlinear terms, in conjunction with the tensorisation coming from the Lie algebra. Numerical tests are provided using a manufactured solution, and show that both schemes display a convergence in -norm of the potential and electrical fields in (provided that the time step is of that order), where is the polynomial degree chosen for the DDR complex. We also numerically demonstrate the preservation of the constraint.
Paper Structure (22 sections, 2 theorems, 58 equations, 6 figures, 3 tables)

This paper contains 22 sections, 2 theorems, 58 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Lie Algebra-valued serendipity Discrete De Rham complex
  3. Mesh notations
  4. Serendipity DDR complex
  5. Spaces and serendipity operators
  6. Operators on the gradient space
  7. Operators on the curl space
  8. Operators on the divergence space
  9. Serendipity DDR complex
  10. Discrete $L^2$-inner products
  11. Lie algebra-valued serendipity DDR complex
  12. Two DDR-based schemes for the Yang--Mills equations
  13. Weak constrained form of the equations
  14. Schemes
  15. Discrete energy and constraint preservation
  16. ...and 7 more sections

Key Result

Proposition 4

For any choice of $\mathfrak{N}$, if $(\underline{\boldsymbol{A}}_h^n,\underline{\boldsymbol{E}}_h^n,\underline{\lambda}_h^n)$ solve eq:ym.lm.scheme then, for all $\underline{q}_h\in\underline{X}_{\mathop{\mathrm{\bf grad}}\nolimits,h}^{k,\mathfrak{g}}$, the quantity $\mathfrak C^n(\underline{q}_h)$

Figures (6)

  • Figure 1: "Voro-small-0" mesh
  • Figure 2: "Tetgen-Cube-0" mesh
  • Figure 3: "Cubic-Cells" mesh
  • Figure 5: "Voro-small-0" mesh
  • Figure 6: "Tetgen-Cube-0" mesh
  • ...and 1 more figures

Theorems & Definitions (8)

  • Remark 1: Approximation properties of the potential reconstructions in $\underline{X}_{\mathop{\mathrm{\bf grad}}\nolimits,T}^{k}$
  • Remark 2: Motivation for the discretisation of the nonlinear terms
  • Remark 3: Discretisation of the linear terms
  • Proposition 4: Constraint preservation
  • proof
  • Proposition 5: Energy dissipation
  • proof
  • Remark 6: Convergence results for $\underline{\lambda}_h$