Project A4 • Time integration of Maxwell and wave-type equations

In 1999/2000, T. Namiki, F. Zheng, Z. Chen and J. Zhangwe proposed such a splitting for the linear (3D) Maxwell equations \begin{alignat*}{2} \partial_t \mathbf{H} &= - \mu^{-1} \operatorname{curl} \mathbf{E}, \\ \partial_t \mathbf{E} &= \hphantom{-} \varepsilon^{-1} \operatorname{curl} \mathbf{H} - \varepsilon^{-1}\mathbf{J}, \qquad \qquad \\ \operatorname{div} (\mathbf{\mu H}) &= 0, \\ \operatorname{div} (\mathbf{\varepsilon E}) &= \rho, \end{alignat*} in a finite difference framework for constant electric permittivity $ \varepsilon $ and magnetic permeability $ \mu $. Time integration is performed using the Peaceman-Rachford scheme, which is unconditionally stable and of conventional order two in time.

In this part of the project, we analyzed this scheme and several variants of it, and provided rigorous error bounds for the semidiscretization in time. In addition, we studied qualitative properties like the preservation of divergence conditions, energy conservation, and decay for large times, both for the exact solution and its numerical approximation, see [Eil17, RS18, EJS19, Zer20a].

We also combined the Peaceman-Rachford scheme with a discontinuous Galerkin spatial discretization and obtained error bounds in a fully discrete setting (see also the section about Time integration for Friedrichs' systems).

Moreover, we also investigated the situation of discontinuous material parameters $ \varepsilon, \mu $ in [Zer20b, Zer22b, ZJ22]. In [DZ22] we investigated the special situation where the piecewise constant coefficients have a jump on a surface in the middle of the cuboid. Continue reading. Collapse content.

Related software can be found in the project ADI Maxwell.

These local time integrators have in common, that the leapfrog scheme is applied on the coarse elements (which is the majority of the elements). Two basic ideas for the remaining elements are to use either an implicit or an explicit scheme with a more favorable CFL condition or smaller step sizes. If we use an implicit scheme we call it locally implicit. The other choice describes a local time-stepping method.

For both, locally implicit methods and local time-stepping methods (LTS) we provided rigorous error bounds under a CFL condition which is independent of the fine mesh elements.

Maxwell equations (Locally implicit methods (LIM))

We considered locally implicit methods where the Crank--Nicolson method is employed on the implicit (fine elements and their direct coarse neighbors) and the leapfrog method on the explicit part (the remaining coarse elements).

For discontinuous Galerkin (dG) space discretizations of Maxwell and wave-type problems, we showed stability under an optimal CFL condition, which only depends on the smallest diameter of the coarse cells. Relying on this result we performed 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 [HS16, HS19]. Since the implicit scheme is unconditionally stable, we were able to show that the CFL condition is restricted only by the coarse elements.

Moreover, in a related approach, we proposed an idea which operates on the numerical linear algebra level within the implementation of a higher-order implicit Runge-Kutta method. More precisely, we suggested to use a preconditioned Krylov subspace method to iteratively solve the linear systems arising in an algebraically stable Runge-Kutta method such as a Gauss collocation method. For the proposed preconditioned Krylov subspace method we rigorously proved that the number of iterations required to reach a certain accuracy is bounded independently of the fine elements in the mesh, cf. [HKK22].

Wave equations (Local time-stepping (LTS) and LIM method)

We constructed and analyzed methods for dG space discretizations of wave equations of the form \begin{align} \label{eq:acousticwave-A4} \partial_{t}^2 u = \mathrm{div} (c^2 \nabla u) + f \qquad \text{in } \Omega \times(0,T], \qquad\quad u(0)=u_0,\quad \partial_t u(0)=v_0, \end{align} on a bounded domain $\Omega$ with wave speed $c$, forcing term $f$, final time $T$, and initial conditions $u_0$, $v_0$.

As preliminary work we constructed and analyzed leapfrog-Chebyshev schemes and a class of multirate schemes for second-order semilinear ODEs, see [CHS20, Car21, CH22].

Based on these results, we presented a rigorous error analysis of local time-stepping methods and locally implicit schemes for linear wave equations \eqref{eq:acousticwave-A4} in [CH23]. For the space discretization we employ a symmetric weighted interior penalty dG method. To ensure stability independent of the fine mesh, one uses techniques based on previous work in [HS16, HS19]. Moreover, we showed optimal convergence order in both space and time.

Implementation details on leapfrog-type multirate methods can be found in the software package Leapfrog-Chebyshev and leapfrog-type multirate methods.
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We proved optimal error bounds for central fluxes dG discretizations of wave-type equations of Friedrichs' type under suitable regularity assumptions of the solution [HK20]. In [HK22], we then combined this dG space discretization with the ADI (Peaceman-Rachford) splitting scheme in time and showed that the resulting approximations as well as discrete derivatives thereof satisfy error bounds of the order of the polynomial degree used in the dG discretization and order two in time.

Moreover, in the book chapter [Dea23], we established a systematic procedure to derive error bounds of dG space discretizations of \eqref{eq:fried-sys} combined with the Crank-Nicolson or the ADI (Peaceman-Rachford) method for the time integration. If the Friedrichs' operator admits a two-field structure, e.g., for Maxwell equations, our analysis also covers the leapfrog and the locally implicit scheme. This nicely merges and considerably generalizes the research conducted in the two doctoral theses [Koe18, Stu17].

TiMaxdG is a related software package and can be found here. The numerical solutions of Friedrichs' systems is also considered in A3 and A7.

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The main difficulty in the numerical treatment of these equations is given, for example in the case of the Westervelt equation, \begin{equation*} \partial_t^2 u = c(u)^2 \Delta u \end{equation*} by the state-dependent wave speed $c$. In this case, it is given by $c^2(u) = \frac{1}{1-2 u}$, and thus the waves travels faster when $u$ close to $\frac12$. This leads to wave fronts as shown in the picture.

Using a framework due to Kato, error bounds for the time discretization of wave and Maxwell equations with certain Runge--Kutta methods were analyzed in [HP17, HPS18] and with exponential integrators in [Doe21, DH22] under reduced regularity assumptions only exploiting regularity which comes from a wellposedness result. For (non-conforming) finite elements, error bounds are shown in [HM22] in a unified setting which allows to treat wave and Maxwell equations simultaneously. This approach was generalized in [Mai20, Mai22] to the full discretization using two variants of the midpoint rule.

A major problem in the spatial discretization is to control the state-dependent (discretized) wave speed which includes pointwise information on the discrete solution. One way to do this, is to use inverse estimates which lead to unsatisfactory restrictions on the ratio of the time step size and the mesh width.

Recently, we have proposed two possible solutions to this problem. For a non-autonomous wave equations, we derived maximum norm error bounds in [DLM21]. Such bounds can be used to circumvent inverse estimates and instead directly bound the numerical solution. This technique was then adapted and extended to quasilinear wave equations in [Doe22], where the CFL conditions could be strongly improved.

For quasilinear Maxwell equations, a new approach was presented in [Sch23] which derives higher-order a priori bounds on the numerical solution first. These bounds are then used to provide rigorous error bounds.

The software package QuasiWave contains implementations of different time integration methods for spatially discretized quasilinear acoustic wave equations. For the analytical questions within this project there is a close collaboration with A5.

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