A Family of Linear Singly Diagonal Runge-Kutta Methods and High Order Pade's Schemes for ODE

Hélène Barucq 1 Marc Durufle 1 Mamadou N'Diaye 1, 2
1 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau], Inria Bordeaux - Sud-Ouest
Abstract : The solution of wave equations with locally time-stepping is of particular interest in case of realistic applications for which local high-order space approximations or locally refined mesh are relevant. The numerical simulation of waves inside the Earth is a challenging example. The general form of the corresponding ODE (Ordinary Differential Equations) reads: $M_h dU/dt + K_h U = F (t$), where $M_h$ is the mass matrix and $K_h$ is the stiffness matrix. The problem is stiff because the eigenvalues of $M_h^{−1} K_h$ may be large. As a consequence, the stability condition of explicit time schemes might become too restrictive. As a preliminary work for the design of high order locally implicit discretizations of wave problems, we provide a performance assessment of different explicit and implicit one-step schemes. The comparison criteria are based on the amplitude and phase errors. Following Burrage's work ([1]) we propose a family of Linear Singly Diagonal Implicit Runge-Kutta schemes (Linear SDIRK) and we analyse the construction of high order schemes by optimizing the constant error. Then we present a new high order non-dissipative time scheme derived from the Padé approximation (see [2]) of an exponential function. Numerical experiments are conducted in $1-D$ and $2-D$ for Maxwell's equations. The numerical results are performed with high order finite elements by using the code Montjoie (see [3]).
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Hélène Barucq, Marc Durufle, Mamadou N'Diaye. A Family of Linear Singly Diagonal Runge-Kutta Methods and High Order Pade's Schemes for ODE. Mathias - annual TOTAL seminar on Mathematics, Numerical simulations, Applied Maths, Numerical Methods, HPC, Parallel Programming, Data processing, Optimization, Oct 2016, Val d'Europe, France. ⟨hal-01406653⟩

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