
A second order finite element method with mass lumping for wave equations in H(div)
We consider the efficient numerical approximation of acoustic wave propa...
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An Application of Rubi: Series Expansion of the Quark Mass Renormalization Group Equation
We highlight how Rulebased Integration (Rubi) is an enhanced method of ...
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A HighOrder LowerTriangular PseudoMass Matrix for Explicit Time Advancement of hp Triangular Finite Element Methods
Explicit time advancement for continuous finite elements requires the in...
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High order integrators obtained by linear combinations of symmetricconjugate compositions
A new family of methods involving complex coefficients for the numerical...
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Optimal and LowMemory NearOptimal Preconditioning of Fully Implicit RungeKutta Schemes for Parabolic PDEs
RungeKutta (RK) schemes, especially GaussLegendre and some other fully...
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On Fake Accuracy Verification
In this paper, we reveal a mechanism behind a fake accuracy verification...
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An exponential integrator/WENO discretization for sonicboom simulation on modern computer hardware
Recently a splitting approach has been presented for the simulation of s...
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Highorder implicit time integration scheme based on Padé expansions
A singlestep highorder implicit time integration scheme for the solution of transient and wave propagation problems is presented. It is constructed from the Padé expansions of the matrix exponential solution of a system of firstorder ordinary differential equations formulated in the statespace. A computationally efficient scheme is developed exploiting the techniques of polynomial factorization and partial fractions of rational functions, and by decoupling the solution for the displacement and velocity vectors. An important feature of the novel algorithm is that no direct inversion of the mass matrix is required. From the diagonal Padé expansion of order M a timestepping scheme of order 2M is developed. Here, each elevation of the accuracy by two orders results in an additional system of real or complex sparse equations to be solved. These systems are comparable in complexity to the standard Newmark method, i.e., the effective system matrix is a linear combination of the static stiffness, damping, and mass matrices. It is shown that the secondorder scheme is equivalent to Newmark's constant average acceleration method, often also referred to as trapezoidal rule. The proposed time integrator has been implemented in MATLAB using the builtin direct linear equation solvers. In this article, numerical examples featuring nearly one million degrees of freedom are presented. Highaccuracy and efficiency in comparison with common secondorder time integration schemes are observed. The MATLABimplementation is available from the authors upon request or from the GitHub repository (to be added).
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