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  1. ; ; ( , Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1)
    Melenk, J.M. ; Perugia, I. ; Schöberl, J. ; Schwab, C (Ed.)
    The matrix valued exponential function can be used for time-stepping numerically stiff discretization, such as the discontinuous Galerkin method but this approach is expensive as the matrix is dense and necessitates global communication. In this paper, we propose a local low-rank approximation to this matrix. The local low-rank construction is motivated by the nature of wave propagation and costs significantly less to apply than full exponentiation. The accuracy of this time stepping method is inherited from the exponential integrator and the local property of it allows parallel implementation. The method is expected to be useful in design and inverse problems where many solves of the PDE are required. We demonstrate the error convergence of the method for the one-dimensional (1D) Maxwell’s equation on a uniform grid. 
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    Accepted Manuscript Full Text Available
  2. ; ; ; ( , Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1)
    Melenk, J.M. ; Perugia, I. ; Schöberl, J. ; Schwab, C. (Ed.)
    Accepted Manuscript Full Text Available
  3. ; ; ; ( , Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018. Lecture Notes in Computational Science and Engineering. Springer.)
    Sherwin, S. ; Moxey, D. ; Peiro, J. ; Vincent, P. ; Schwab, C. (Ed.)
    Runge-Kutta time-stepping methods in general suffer from order reduction: the observed order of convergence may be less than the formal order when applied to certain stiff problems. Order reduction can be avoided by using methods with high stage order. However, diagonally-implicit Runge-Kutta (DIRK) schemes are limited to low stage order. In this paper we explore a weak stage order criterion, which for initial boundary value problems also serves to avoid order reduction, and which is compatible with a DIRK structure. We provide specific DIRK schemes of weak stage order up to 3, and demonstrate their performance in various examples. 
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