[Paper Review] Mimetic Explicit Time Discretiztions

Stanly Steinberg|arXiv (Cornell University)|May 27, 2016

Advanced Numerical Methods in Computational Mathematics27 references3 citations

TL;DR

This paper presents a mimetic finite difference method that combines staggered spatial and explicit time discretizations to solve wave equations with second-order accuracy, energy conservation, and unconditional stability under a time step constraint. The method preserves key continuum properties—such as divergence-free fields and energy-like invariants—by constructing a discrete energy functional that mirrors the continuous system's structure, validated through scalar and Maxwell’s equations in 1D, 2D, and 3D with exact conservation up to machine precision.

ABSTRACT

This paper is part of a program to combine a staggered time and staggered spatial discretization of continuum mechanics problems so that any property of the continuum that is proved using vector calculus can be proven in an analogous way for the discretized system. We require that the discretizations be second order accurate and have a conserved quantity that approximates the energy for the system and guarantees stability for a reasonable constraint on the time step. We also require that the discretization is time explicit so as to avoid the solution of large system of possibly nonlinear algebraic equations. The well known Yee grid discretization of Maxwell's equations is the same as our discretization and is an early example of using a staggered space and time grid . To motivate our discussion we begin by studying the staggered time or leapfrog discretization of the harmonic oscillator and use this to introduce the modification of the energy that is conserved. Next we use systems of linear equations to motivate the definition of the modified energy for more complex systems of ordinary differential equations and then apply our ideas to the scalar wave equation in one spatial dimension. We finish by discretizing the three dimensional scalar wave and Maxwell's equations. Because the spatial discretization is mimetic, we obtain that the divergence of the electric and magnetic fields are constant when there are no sources. Using the mimetic properties the proof of this trivial and is essentially the same as in the continuum.

Motivation & Objective

• To develop a time-explicit, second-order accurate finite difference scheme for wave equations that preserves fundamental continuum properties such as energy conservation and divergence-free fields.
• To construct a discrete energy functional that approximates the continuous energy and remains conserved under time evolution, ensuring numerical stability.
• To extend mimetic spatial discretizations—previously used for elliptic and parabolic problems—to hyperbolic wave systems using staggered space-time grids.
• To enable stable simulations of inhomogeneous and anisotropic materials by combining mimetic spatial operators with leapfrog time stepping.
• To demonstrate that the discrete system inherits the algebraic structure of the continuum, including exact sequence properties and adjoint relationships, via diagram chasing and discrete star operators.

Proposed method

• Use of a staggered time (leapfrog) discretization for first-order systems derived from second-order wave equations, ensuring second-order accuracy and explicit time stepping.
• Construction of a modified discrete energy functional that acts as a conserved quantity, analogous to the continuous energy, and is positive for sufficiently small time steps.
• Employment of two dual grids (primal and dual) where nodes of one grid coincide with cell centers of the other, enabling mimetic spatial operators for gradient, curl, and divergence.
• Definition of discrete star operators (multiplication operators) that map between primal and dual grids, preserving duality and enabling consistent inner products.
• Application of discrete exact sequences to ensure that mimetic properties—such as div(curl) = 0 and the compatibility of operators—are preserved at the discrete level.
• Implementation of boundary conditions via mixed (Robin-type) conditions, with corner point values handled consistently to avoid programming inconsistencies.

Experimental results

Research questions

• RQ1 Can a time-explicit, second-order accurate finite difference scheme be constructed for wave equations that conserves a discrete energy functional analogous to the continuous energy?
• RQ2 How can mimetic spatial discretizations—previously used for elliptic and parabolic problems—be extended to hyperbolic wave systems while preserving key continuum properties?
• RQ3 What is the structure of the discrete energy functional that ensures stability and second-order accuracy in time for wave equations on staggered grids?
• RQ4 How do discrete differential operators (gradient, divergence, curl) and their adjoints behave under mimetic discretization, and what algebraic properties do they preserve?
• RQ5 To what extent can the method be generalized to inhomogeneous and anisotropic materials, and how is this reflected in the discrete operators and energy functional?

Key findings

• The discrete energy functional is conserved to at least 1 part in 10^15 in numerical tests, confirming exact conservation in practice.
• The method achieves second-order accuracy in time and space for the 2D scalar wave equation, with some cases showing fourth-order accuracy or exact solutions when parameters are tuned.
• The mimetic spatial discretization ensures that the divergence of the electric and magnetic fields remains zero in the absence of sources, just as in the continuum.
• The discrete system inherits the exact sequence structure of the continuum, with discrete operators satisfying the same algebraic identities (e.g., div(curl) = 0) due to diagram chasing and proper star operator definition.
• The use of dual grids and discrete star operators enables consistent handling of anisotropic and inhomogeneous material properties in the wave equation.
• Boundary conditions are implemented via mixed (Robin-type) forms, and corner point values are handled without affecting the solution in explicit time-stepping schemes.