Multi-level explicit local time-stepping methods for second-order wave equations
Introduction
Second-order wave equations are ubiquitous across a wide range of applications from acoustics, electromagnetics, and elasticity. Their spatial discretization by standard finite difference or finite element methods typically leads to a large system of second-order ordinary differential equations. When explicit time integration is subsequently used, the time-step will be governed by the smallest elements in the mesh for numerical stability. Near corners, material interfaces or other small-scale geometric features, adaptive mesh refinement is certainly key for the accurate simulation of wave phenomena [1]. Local mesh refinement, however, severely impedes the efficiency of explicit time-marching methods because of the overly small time-step dictated by but a few tiny elements. When mesh refinement is restricted to a small portion of the computational domain, the use of implicit methods or a small time-step everywhere, is rather high a price to pay.
Local time-stepping (LTS) methods overcome the bottleneck due to local refinement by dividing the mesh into two distinct regions: the “coarse” region, which contains the larger elements and is integrated in time using an explicit method, and the “fine” region, which contains the smaller elements and is integrated in time using either smaller time-steps or an implicit scheme.
Locally implicit methods build on the long tradition of hybrid implicit–explicit (IMEX) algorithms for operator splitting in computational fluid dynamics—see [2], [3] and the references therein. Here, a linear system needs to be solved inside the refined region at every time-step, which becomes not only increasingly expensive with decreasing mesh size, but also increasingly ill-conditioned as the grid-induced stiffness increases [4]. Moreover, even when each individual method has order two, the implicit–explicit component splitting can reduce by one the overall space–time convergence rate of the resulting scheme [5], [6]. Recently, Descombes, Lanteri and Moya [6] remedied that unexpected loss in accuracy and hence recovered second-order convergence, by using the LF/CN-IMEX approach of Verwer [7] instead, yet at the price of a significantly larger albeit sparse linear system.
In contrast, locally explicit time-stepping methods remain fully explicit by taking smaller time-steps in the “fine” region, that is precisely where the smaller elements are located. In the mid- to late 80s, Berger and Oliger [8] and Berger and Colella [9] proposed a space–time adaptive mesh refinement (AMR) strategy for nonlinear hyperbolic conservation laws. Based on a hierarchy of rectangular finite-difference grids, it was later extended to hyperbolic equations not necessarily in conservation form by using wave propagation algorithms [10]. Higher accuracy was achieved more recently by combining the AMR approach with weighted essentially non-oscillatory (WENO) reconstruction techniques [11], [12].
Because they easily accommodate unstructured meshes, finite element methods (FEM) are usually more effective in the presence of complex geometry or adaptive mesh refinement. For hyperbolic conservation laws, discontinuous Galerkin (DG) FEM are particularly well-suited because they are locally conservative. In [13], Flaherty et al. proposed probably the first local time-stepping (LTS) strategy for a DG-FEM, where each element selects its time-step according to the local Courant–Friedrichs–Lewy (CFL) stability condition. By using the Cauchy–Kovalevskaya procedure within each element, arbitrary high-order (ADER) DG schemes achieve high-order accuracy both in space and time [14] and also permit each element to use its optimal time-step determined by the local stability condition. They were also successfully applied to electromagnetic [15] and elastic wave propagation [16].
The standard method of lines approach leaves much flexibility in the choice of the spatial discretization, as it applies not only to DG but also to continuous (conforming) FE or even finite difference methods. Local time-stepping methods then integrate the resulting system of ODEs by taking larger time-steps for larger elements, thus concentrating work on the smaller ones. In [17], a local time-stepping scheme based on a second-order Runge–Kutta method was proposed for nonlinear conservation laws. Also known as multirate or multiple time-stepping methods in the ODE literature [18], various high order LTS methods have been proposed for numerical wave propagation based on classical Adams–Bashforth multistep methods [19]; they can also be interpreted as particular approximations of exponential-Adams multi-step methods [20]. Recently, Runge–Kutta based explicit LTS of arbitrarily high-order were proposed for wave propagation in [21].
In the absence of forcing and dissipation, the classical wave equation conserves the total energy. When a symmetric spatial FD or FE discretization is combined with a centered time-marching scheme, such as the standard leap-frog (LF) (also known as Newmark or Störmer–Verlet) method, the resulting fully discrete formulation will also conserve (a discrete version of) the energy. Highly efficient in practice, centered time discretizations also display remarkably high accuracy over long times and remain even nowadays probably the most popular methods for the time integration of wave equations. In [22] Collino, Fouquet and Joly proposed an LTS method for the wave equation in first-order form, which conserves a discrete energy yet requires every time-step the solution of a linear system on the interface between the coarse and the fine mesh. It was analyzed in [23], [24] and later extended to elastodynamics [25] and Maxwell’s equations [26]. By combining a symplectic integrator with a DG discretization of Maxwell’s equations in first-order form, Piperno [27] proposed a second-order explicit local time-stepping scheme, which also conserves a discrete energy. Starting from the standard LF method, the authors proposed energy conserving fully explicit LTS integrators of arbitrarily high accuracy for the wave equation [28]; that approach was extended to Maxwell’s equations in [29]. An -version, where not only the time-step but also the order of approximation is adapted within different regions of the mesh, was proposed in [30] and later applied to a realistic geological model [31].
When a region of local refinement itself contains sub-regions of further refinement, those “very fine” elements yet again will dictate the time-step, albeit local, to the entire “fine” region. Then, it becomes more efficient to let the time-marching strategy mimic the multilevel hierarchy of the mesh organized into tiers of “coarse”, “fine”, “very fine”, etc. elements by introducing a corresponding hierarchy into the time-stepping method. Hence, the resulting multi-level local time-stepping (MLTS) method will advance in time by using within each tier of equally sized elements the corresponding optimal time-step.
The outline of our paper is as follows. Starting from a semi-discrete Galerkin finite element formulation of the wave equation, we derive in Section 2 local time-stepping (LTS) methods of arbitrarily high order based on the leap-frog (LF) method; we also recall some of their key properties from [28]. Although first presented in [28], the present derivation is different and crucial for the derivation of the multi-level local time-stepping (MLTS) methods in Section 3. In Section 4, we prove that the second-order MLTS method conserves a discrete energy regardless of the number of intermediate levels. Finally, in Section 5, we present numerical experiments in one and two space dimensions which illustrate the stability and convergence properties of these MLTS schemes.
Section snippets
Local time-stepping
We consider the acoustic wave equation a standard model for second-order hyperbolic problems. Here is a bounded domain in , whereas and are prescribed initial conditions. For simplicity, we impose homogeneous Dirichlet conditions at the boundary, , and assume that is source-free. The density, , and the bulk modulus, , are piecewise smooth, strictly positive and bounded, and hence so is the wave
Multilevel local time-stepping
If the refined part of the mesh itself contains a small subregion of even further local space refinement, it becomes more efficient to introduce yet another level of local time-stepping associated with it. Thus, we let denote the diagonal partitioning matrix whose diagonal entries, equal to zero or one, identify the unknowns associated with the first level of local mesh refinement. Similarly, we let denote the diagonal partitioning matrix associated with the second level of local mesh
Energy conservation
In [28], we proved that both the second-order and the fourth-order two-level LTS methods conserve a discrete energy — see Proposition 2.1, Proposition 2.2. To extend that analysis to an arbitrary number of levels , we first rewrite the algorithm (32) in LF fashion: Proposition 4.1 The algorithm (32) is equivalent towhere is a symmetric matrix.
The proof follows directly from the next two lemmas, which are proved in the Appendix.
Lemma 4.1 Let
Numerical results
We shall now present numerical experiments that confirm the expected order of convergence and demonstrate the versatility of the multilevel local time-stepping (MLTS) methods from Section 3. First, we consider a simple one-dimensional test problem to show that the different MLTS schemes are stable and indeed yield the expected overall rate of convergence when combined with a spatial finite element discretization of comparable accuracy. Then, we consider wave propagation in two space dimensions
Concluding remarks
Starting from the local time-stepping (LTS) methods in [28], we have derived multi-level local time-stepping methods (MLTS) of arbitrarily high order for second-order wave equations. When the elements of the underlying mesh are naturally organized into tiers of “coarse”, “fine”, “very fine”, etc. elements, our MLTS methods apply the same multi-level structure to the time-stepping, without sacrificing accuracy or explicitness. Hence they permit inside every tier of like-sized elements the use
References (42)
- et al.
Application of implicit–explicit high order Runge–Kutta methods to discontinuous-Galerkin schemes
J. Comput. Phys.
(2007) - et al.
Locally implicit discontinuous Galerkin method for time domain electromagnetics
J. Comput. Phys.
(2010) - et al.
Adaptive mesh refinement for hyperbolic partial differential equations
J.~Comput.~Phys.
(1984) - et al.
Local adaptive mesh refinement for shock hydrodynamics
J.~Comput.~Phys.
(1989) - et al.
ADER-WENO finite volume schemes with space–time adaptive mesh refinement
J.~Comput.~Phys.
(2013) - et al.
Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws
J. Parallel Distrib. Comput.
(1997) An efficient local time-stepping scheme for solution of nonlinear conservation laws
J.~Comput.~Phys.
(2010)- et al.
High-order explicit local time-stepping methods for damped wave equations
J. Comput. Appl. Math.
(2013) - et al.
Space–time mesh refinement for elastodynamics. Numerical results
Comput. Methods Appl. Mech. Engrg.
(2005) - et al.
Conservative space–time mesh refinement methods for the FDTD solution of Maxwell’s equations
J. Comput. Phys.
(2006)
Explicit local time-stepping methods for Maxwell’s equations
J. Comput. Appl. Math.
Finite Elements with Mesh Refinement for Wave Equations in Polygons, Tech. Rep., Seminar for Applied Mathematics
Numerical approximations to nonlinear conservation laws with locally varying time and space grids
Math. Comp.
Implicit–explicit methods for time-dependent partial differential equations
SIAM J. Numer. Anal.
Locally implicit time integration strategies in a discontinuous Galerkin method for Maxwell’s equations
J. Sci. Comput.
Component splitting for semi-discrete Maxwell equations
BIT
Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems
SIAM J.~Numer.~Anal.
Adaptive mesh refinement techniques for high-order shock capturing schemes for multi-dimensional hydrodynamic simulations
Internat. J. Numer. Methods Fluids
A discontinuous Galerkin scheme based on a space–time expansion. I. Inviscid compressible flow in one space dimension
J. Sci. Comput.
A high-order discontinuous Galerkin method with local time stepping for the Maxwell equations
Int. J. Numer. Model.
An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes -V. Local time stepping and p-adaptivity
Geophys. J. Int.
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