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

Principal investigators

  Prof. Dr. Marlis Hochbruck (7/2015 - )
  Prof. Dr. Tobias Jahnke (7/2015 - 6/2019)
  Prof. Dr. Roland Schnaubelt (7/2015 - )

Project summary

Efficiently solving wave-type equations such as the time-dependent Maxwell equations or the acoustic wave equation on different type of space domains is a crucial and challenging task in numerous applications.

In practice, classical explicit methods such as the well-known Yee scheme for the Maxwell equations are often used to approximate the solution. However, such methods suffer from numerical instabilities, which can only be avoided by choosing a sufficiently small time-step size. This can severely affect the performance of the scheme. On the other hand, traditional unconditionally stable schemes are implicit and thus performing one step of the scheme can be rather expensive.

Fortunately, there are various schemes with favorable stability properties of implicit methods which can nevertheless be executed efficiently. Prominent examples include alternating direction implicit (ADI) methods on rectangular or cuboidal domains, and local time integrators if only a small part of the grid contains tiny elements that cause instabilities in fully explicit methods.

Within this project, we also study quasilinear equations, which impose more theoretical questions. One 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 coefficients.



Alternating direction implicit (ADI) methods for linear Maxwell equations

Alternating direction implicit (ADI) methods are a class of geometrical splitting methods, where the spatial differential operator is split in a way that the resulting split operators lead to essentially one-dimensional problems. This allows the efficient implementation of implicit, unconditionally stable time integration schemes.

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).

Left: Sparsity pattern of the coefficient matrix belonging to linear systems that have to be solved in each time step in the Peaceman–Rachford scheme
Right: Example for the ordering used to obtain an efficient scheme.

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.

Example of a heterogeneous model domain with discontinuous material parameters, see also [ZJ22].

Related software can be found in the project ADI Maxwell.

Local time integrators for linear Maxwell and wave-type equations

By far the most popular time integration method for spatially discretized wave-type equations, such as Maxwell or acoustic wave equations, is the leapfrog (Störmer–Verlet) method. It is easy to implement and very efficient. The main drawback of this scheme is its restriction to small time-step sizes (CFL condition).

If the number of tiny elements in the spatial discretization is rather small compared to the total number of elements, local time integrators provide an efficient alternative for the time integration.

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.

Examples of locally refined grids where the number of tiny elements is small compared to the number of coarse elements. Pictures taken from [AM17] and [Stu17, Figure 5.3].
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.

Time integration for Friedrichs' systems

The class of Friedrichs' systems, given by \begin{equation} \label{eq:fried-sys} \begin{aligned} \partial_t u & = M^{-1}L u + f && \text{on } \mathbb{R}_+ \times \Omega,\\ \ u(0) & = u_0 && \text{on } \Omega, \end{aligned} \end{equation} with a material tensor $ M $ and the Friedrichs' operator $ L $, comprises linear Maxwell, advection, and wave equations. We established a systematic procedure to derive error bounds of discontinuous Galerkin (dG) space discretizations combined with the Crank-Nicolson, the leapfrog, the locally implicit, and the ADI (Peaceman-Rachford) scheme for the time integration for a general class of Friedrichs' systems.

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.

Nonlinear Maxwell and wave-type equations

Nonlinear Maxwell and wave-type equations are an important tool in modeling physical phenomena such as nonlinear acoustics and optics. In this project, we discuss the space, time, and full discretization and establish error bounds in different scenarios.

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.


  1. . Strong norm error bounds for quasilinear wave equations under weak CFL-type conditions. Found. Comput. Math., 48pp., February . URL Online first. [preprint] [files] [bibtex]

  2. and . Analysis of a Peaceman–Rachford ADI scheme for Maxwell equations in heterogenous media. J. Math. Anal. Appl., 527(1, Part 1):127355, November . URL [preprint] [bibtex]

  3. , , and . Maximum norm error bounds for the full discretization of nonautonomous wave equations. IMA J. Numer. Anal., drad065, September . URL [preprint] [files] [bibtex]

  4. and . Wellposedness and regularity for linear Maxwell equations with surface current. Z. Angew. Math. Phys., 74(4):131, August . URL [preprint] [bibtex]

  5. , , , , , and . Wave Phenomena. Mathematical Analysis and Numerical Approximation, volume 49 of Oberwolfach Seminars. Birkhäuser Cham, March . [bibtex]

  6. , , and . Optimal $W^{1,\infty}$-estimates for an isoparametric finite element discretization of elliptic boundary value problems. Electron. Trans. Numer. Anal., 58:1–21, January . URL [preprint] [bibtex]

  7. . Analysis of a dimension splitting scheme for Maxwell equations with low regularity in heterogeneous media. J. Evol. Equ., 22(4):90–135, November . URL [preprint] [bibtex]

  8. and . Error analysis of multirate leapfrog-type methods for second-order semilinear ODEs. SIAM J. Numer. Anal., 60(5):2897–2924, October . URL [preprint] [files] [bibtex]

  9. and . Error analysis for space discretizations of quasilinear wave-type equations. IMA J. Numer. Anal., 42(3):1963–1990, July . URL [preprint] [files] [bibtex]

  10. and . Exponential integrators for quasilinear wave-type equations. SIAM J. Numer. Anal., 60(3):1472–1493, June . URL [preprint] [files] [bibtex]

  11. . Interpolation of a regular subspace complementing the span of a radially singular function. Studia Math., 265(2):197–210, May . URL [preprint] [bibtex]

  12. . Error analysis for full discretizations of quasilinear wave-type equations with two variants of the implicit midpoint rule. IMA J. Numer. Anal., drac010, May . URL [preprint] [bibtex]

  13. and . Error analysis of a fully discrete discontinuous Galerkin alternating direction implicit discretization of a class of linear wave-type problems. Numer. Math., 150(3):893–927, March . URL [preprint] [files] [bibtex]

  14. . A uniformly exponentially stable ADI scheme for Maxwell equations. J. Math. Anal. Appl., 492(1):124442, December . URL [preprint] [bibtex]

  15. and . Error analysis of discontinuous Galerkin discretizations of a class of linear wave-type problems. In W. Dörfler, M. Hochbruck, D. Hundertmark, W. Reichel, A. Rieder, R. Schnaubelt, and B. Schörkhuber, editors, Mathematics of Wave Phenomena, Trends in Mathematics, pages 197–218, October . Birkhäuser Basel. [bibtex]

  16. , , and . On leapfrog-Chebyshev schemes. SIAM J. Numer. Anal., 58(4):2404–2433, August . URL [preprint] [bibtex]

  17. , , and . Error analysis of an energy preserving ADI splitting scheme for the Maxwell equations. SIAM J. Numer. Anal., 57(3):1036–1057, May . URL [preprint] [bibtex]

  18. and . Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell's equations. Math. Comp., 88(317):1121–1153, May . URL [preprint] [bibtex]

  19. and . On the efficiency of the Peaceman–Rachford ADI-dG method for wave-type problems. In F. A. Radu, K. Kumar, I. Berre, J. M. Nordbotten, and I. S. Pop, editors, Numerical Mathematics and Advanced Applications ENUMATH 2017, volume 126 of Lecture Notes in Computational Science and Engineering, pages 135–144, January . Springer International Publishing. [preprint] [bibtex]

  20. and . Error analysis of an ADI splitting scheme for the inhomogeneous Maxwell equations. Discrete Contin. Dyn. Syst., 38(11):5685–5709, November . URL [preprint] [bibtex]

  21. , , and . Convergence analysis of energy conserving explicit local time-stepping methods for the wave equation. SIAM J. Numer. Anal., 56(2):994–1021, April . URL [preprint] [bibtex]

  22. , , and . Error analysis of implicit Runge–Kutta methods for quasilinear hyperbolic evolution equations. Numer. Math., 138(3):557–579, March . URL [preprint] [bibtex]

  23. and . Multilevel local time-stepping methods of Runge–Kutta-type for wave equations. SIAM J. Sci. Comput., 39(5):A2020–A2048, September . URL [preprint] [bibtex]

  24. and . Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations. Numer. Math., 135(2):547–569, February . URL [preprint] [bibtex]

  25. and . Error analysis of a second-order locally implicit method for linear Maxwell's equations. SIAM J. Numer. Anal., 54(5):3167–3191, October . URL [preprint] [bibtex]


  1. and . Robust fully discrete error bounds for the Kuznetsov equations in the inviscid limit. CRC 1173 Preprint 2024/1, Karlsruhe Institute of Technology, January . [files] [bibtex]

  2. and . Error analysis of the Lie splitting for semilinear wave equations with finite-energy solutions. CRC 1173 Preprint 2023/24, Karlsruhe Institute of Technology, November . [bibtex]

  3. . Error analysis of the implicit Euler scheme for the Maxwell–Kerr system. CRC 1173 Preprint 2023/3, Karlsruhe Institute of Technology, January . [bibtex]

  4. and . Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations. CRC 1173 Preprint 2023/2, Karlsruhe Institute of Technology, January . Revised version from May 2023. To be published in Math. Comp. [files] [bibtex]

  5. , , and . Preconditioned implicit time integration schemes for Maxwell's equations on locally refined grids. CRC 1173 Preprint 2022/29, Karlsruhe Institute of Technology, June . [files] [bibtex]


  1. . On leapfrog-Chebyshev schemes for second-order diffferential equations. PhD thesis, Karlsruhe Institute of Technology, December . [bibtex]

  2. . Error analysis of exponential integrators for nonlinear wave-type equations. PhD thesis, Karlsruhe Institute of Technology (KIT), February . [bibtex]

  3. . ADI schemes for the time integration of Maxwell equations. PhD thesis, Karlsruhe Institute of Technology, December . [bibtex]

  4. . Error analysis for space and time discretizations of quasilinear wave-type equations. PhD thesis, Karlsruhe Institute of Technology, May . [bibtex]

  5. . The Peaceman–Rachford ADI-dG method for linear wave-type problems. PhD thesis, Karlsruhe Institute of Technology (KIT), September . [bibtex]

  6. . Error analysis of splitting methods for wave type equations. PhD thesis, Karlsruhe Institute of Technology (KIT), July . [bibtex]

  7. . Locally implicit time integration for linear Maxwell's equations. PhD thesis, Karlsruhe Institute of Technology (KIT), April . [bibtex]

Former staff
Name Title Position
Dr. Postdoctoral and doctoral researcher
Dr. Postdoctoral and doctoral researcher
Prof. Dr. Board member
Dr. Postdoctoral and doctoral researcher
Ph. D. Postdoctoral researcher
Dr. Postdoctoral and doctoral researcher
Dr. Postdoctoral researcher
Dr. Postdoctoral and doctoral researcher
Dr. Postdoctoral and doctoral researcher