Explicit Finite Element Methods for Linear Hyperbolic Systems

Falk, Richard S.; Richter, Gerard R.

doi:10.1007/978-3-642-59721-3_15

Cited by 6 publications

(6 citation statements)

References 11 publications

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“…An innovative implicit space–time DGM with basis functions that are free‐space solutions of the PDE governing the problem of interest was proposed in for the solution of wave propagation problems in the time domain. By partitioning the space–time domain carefully, systems of equations, which are independent of each other, can be formulated on clusters of neighboring elements, independently of other such clusters, yielding a semi‐implicit method .…”

Section: Introductionmentioning

confidence: 99%

A hybrid discontinuous in space and time Galerkin method for wave propagation problems

Wang

Tezaur

Farhat

2014

Numerical Meth Engineering

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SUMMARYMedium-frequency regime and multi-scale wave propagation problems have been a subject of active research in computational acoustics recently. New techniques have attempted to overcome the limitations of existing discretization methods that tend to suffer from dispersion. One such technique, the discontinuous enrichment method, incorporates features of the governing partial differential equation in the approximation, in particular, the solutions of the homogeneous form of the equation. Here, based on this concept and by extension of a conventional space-time finite element method, a hybrid discontinuous Galerkin method (DGM) for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions is described. The discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation. In this contribution, within each finite element, the solutions in the form of polynomial waves are employed. The continuity of these polynomial waves is weakly enforced through suitably chosen Lagrange multipliers. Results for two-dimensional and threedimensional problems, in both low-frequency and medium-frequency regimes, show that the proposed DGM outperforms the conventional space-time finite element method.

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Section: Introductionmentioning

confidence: 99%

A hybrid discontinuous in space and time Galerkin method for wave propagation problems

Wang

Tezaur

Farhat

2014

Numerical Meth Engineering

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“…Various DG methods can be applied to solve linear hyperbolic systems. We mention the classical Runge-Kutta DG method of Cockburn and Shu [10], the nodal DG method of Hesthaven and Warburton [16], the space-time DG method of Falk and Richter [12] and Monk and Richter [23]. All these DG methods use approximate/exact Riemann solvers to define the numerical flux, and are dissipative by design.…”

Section: Introductionmentioning

confidence: 99%

“…12: long time simulation: advection of a Gaussian pulse. We consider the advection equation (4.2) on the unit square with periodic boundary condition and initial condition u(x, y, 0) = exp(−200((x − 0.5) 2 + (y − .5) 2 )).…”

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confidence: 99%

Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems

Shu

2019

Journal of Computational Physics

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We propose energy-conserving discontinuous Galerkin (DG) methods for symmetric linear hyperbolic systems on general unstructured meshes. Optimal a priori error estimates of order k + 1 are obtained for the semidiscrete scheme in one dimension, and in multi-dimensions on Cartesian meshes when tensor-product polynomials of degree k are used. A high-order energyconserving Lax-Wendroff time discretization is also presented.Extensive numerical results in one dimension, and two dimensions on both rectangular and triangular meshes are presented to support the theoretical findings and to assess the new methods. One particular method (with the doubling of unknowns) is found to be optimally convergent on triangular meshes for all the examples considered in this paper. The method is also compared with the classical (dissipative) upwinding DG method and (conservative) DG method with a central flux. It is numerically observed for the new method to have a superior performance for long-time simulations.2010 Mathematics Subject Classification. Primary 65M60, 65M12, 65M15.

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“…Recently, new explicit finite element methods with unstructured meshes based on the discontinuous Galerkin method are suggested for elastodynamics problems [16,17,18]. The methods allow an element-by-element solution in a space-time domain.…”

Section: Introductionmentioning

confidence: 99%

Application of Space-Time Finite Elements to Elastodynamics Problems

Idesman¹

2004

AIP Conference Proceedings

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A nodal discontinuous Galerkin finite element method for nonlinear elastic wave propagation A generalized approach for efficient finite element modeling of elastodynamic scattering in two and three dimensions Abstract. Currently, commercial finite element codes for elastodynamics problems use a finite difference approximation in time and a finite element approximation in space. There are two disadvantages to existing methods: 1) extremely small integration time steps (e.g., nanoseconds) are necessary due to the Courant limit and 2) it is practically impossible to use different time steps for different spatial finite elements. This leads to the significant increase in computing time that could be crucial for time-consuming calculations for real world elastodynamics problems. A new approach with space-time finite elements for the solution of linear elastodynamics problems is suggested. Weak formulations for elastodynamics are based on continuous and discontinuous Galerkin methods. Using unstructured finite element meshes for space-time domains allows an adaptive discretization simultaneously in space and time, i.e. a concentration of unknowns of a discrete model around small space-time domains (where necessary) without a significant increase in a total number of unknowns for the whole problem. The so-called 'subcycling' procedure, i.e. different time increments for different space domains, is automatically included. Also specific 'inverse' dynamics problems (when initial displacements and initial velocities are specified at different time) can be easily solved by the proposed technique. Our initial numerical results for one-dimensional elastodynamics problems have demonstrated the feasibility of implementation of this new numerical method.

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Explicit Finite Element Methods for Linear Hyperbolic Systems

Cited by 6 publications

References 11 publications

A hybrid discontinuous in space and time Galerkin method for wave propagation problems

A hybrid discontinuous in space and time Galerkin method for wave propagation problems

Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems

Application of Space-Time Finite Elements to Elastodynamics Problems

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