Project A4 • Time integration of Maxwell equations

Efficient methods for linear Maxwell equations

It often suffices to consider a linear relation between the electric field \(\mathbf{E}\) and the electric displacement \(\mathbf{D}\), and between the magnetic field \(\mathbf{H}\) and the magnetic flux density \(\mathbf{B}\), namely \[\begin{eqnarray*} \mathbf{D} & = & \varepsilon \mathbf{E}, \\ \mathbf{B} & = & \mu \mathbf{H}, \end{eqnarray*}\] leading to the linear Maxwell equations.

We investigate three prominent examples of efficient methods for these equations, namely alternating direction implicit (ADI), locally implicit and local time-stepping schemes. In most cases the error analysis of these schemes is reduced to formal arguments based on Taylor expansions. These results have to be treated with caution because they either require very regular solutions or yield error estimates involving the norm of the space discretization matrices, which might become arbitrarily large when the spatial discretization is refined.

In this project, our main goal is to show rigorous error bounds, which are robust under mesh refinement and hold for moderate or even low regularity assumptions on initial values and coeffcients. We further study qualitative properties like preservation of divergence conditions or decay for large times, both for the exact solution and its numerical approximation.

Alternating direction implicit (ADI) method

ADI methods are a class of geometrical splitting methods, where the spatial differential operator is split so that the resulting split operators lead to one-dimensional problems. This allows for implicit, unconditionally stable time integration schemes which can be implemented extremely efficient.

In particular, we investigate the splitting for the linear (3D) Maxwell equations which was proposed for a finite difference framework by T. Namiki, and F. Zheng, Z. Chen and J. Zhang in 1999/2000. Time integration is performed using the Peaceman–Rachford scheme, which is unconditionally stable and of conventional order two.

Our aim is to deduce rigorous error estimates. Therefore, we combine this splitting with a central flux discontinuous Galerkin (dG) spatial discretization. For this fully discrete scheme, the efficiency is not as straight-forward as it is for the finite difference case. But we show that if a tensor product grid and suitable ordering of the degrees of freedom are used, the ADI-dG scheme can still be performed in optimal (linear) complexity. Further, we show optimal convergence rates for the fully discrete scheme, namely order two in time and order \(k\) in space, where \(k\) is the polynomial degree of the spatial ansatz functions.

Sparsity pattern of the coefficient matrix belonging to linear systems that have to be solved in each timestep.

Example for the ordering used to obtain an efficient scheme.

The main caveat of ADI methods is, however, that they can only be implemented efficiently if the geometry is restricted to (unions of) cuboids and if the spatial discretization is performed on a tensor product grid.

Methods for locally refined grids

For linear Maxwell equations on domains with a more general geometry, discontinuous Galerkin or finite element methods on unstructured grids are methods of choice for the space discretization. If the number of tiny elements is rather small compared to the total number of elements, locally implicit or local time-stepping methods provide an efficient alternative for the time integration.

Example of a locally refined grid.

The basic idea for locally implicit methods is to treat only the tiny elements and their direct neighbors implicitly, whereas the remaining (coarse) elements are treated by an explicit method. Since the implicit scheme is unconditionally stable, the CFL condition is restricted only by the coarse elements.

Solution of linear Maxwell equations.

In particular, we employ the Crank–Nicolson method for the implicit and the leapfrog method for the explicit part and a space discretization with either central or upwind flux discontinuous Galerkin method.

For either central or upwind flux discontinuous Galerkin space discretization, we show stability under an optimal CFL condition, which only depends on the smallest diameter of the coarse cells. Relying on this result we perform a rigorous error analysis, which shows that the convergence rates are the same as for the fully implicit or explicit schemes: The scheme is of quadratic order in time and spatial order \(k\) for central and \(k+1/2\) for upwind fluxes.

A related idea motivates local time-stepping methods. For the tiny elements an explicit time integration scheme with a small time step size is used, for the other part with the large cells the same explicit scheme with a larger time step size.

For local time-stepping schemes, however, a rigorous stability and error analysis is not done yet.

Quasilinear Maxwell equation

In many applications one has to use nonlinear material laws \(D = \epsilon(E) E\) and \(B = \mu(H) H\) which lead to nonlinear Maxwell equations. Unfortunately, the analysis of these equations is much more involved than in the linear case and results concerning well-posedness and regularity of solutions are scarce.

To gain more insight into the analysis of numerical methods for these equations, in a first step, we focus on algebraically stable and coercive Runge-Kutta methods, such as Gauß and Radau collocation methods. Based on a refinement of the analytical setting of Kato's well-posedness theory we show rigorous error estimates for a class of nonlinear material laws.

However, this approach is restricted to certain cases like full space problems or quasilinear wave equations with Dirichlet boundary conditions. To tackle the standard boundary conditions of a perfect conductor for the quasilinear Maxwell system, for instance, one has to change the analytical framework and use recent results from project A5 of our CRC. This is subject of on-going work.

So far we have treated the time integration scheme on the level of the evolution equation. In a next step we will analyze the full discretization using e.g. the discontinuous Galerkin method in space. Continue reading. Collapse content.