Higher-order finite elements with mass-lumping for the 1D wave equation
Finite Elements in Analysis and Design - Special issue: selection of papers presented at ICOSAHOM'92
Numerical methods for ordinary differential equations in the 20th century
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation
SIAM Journal on Numerical Analysis
ADER: Arbitrary High Order Godunov Approach
Journal of Scientific Computing
Explicit Runge-Kutta residual distribution schemes for time dependent problems: Second order case
Journal of Computational Physics
Mesh-dependent stability for finite element approximations of parabolic equations with mass lumping
Journal of Computational and Applied Mathematics
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Computers are becoming sufficiently powerful to permit to numerically solve problems such as the wave equation with high-order methods. In this article we will consider Lagrange finite elements of order k and show how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. We compare two high-order time discretizations: an explicit one using a Taylor expansion in time (a Cauchy-Kowalewski procedure) and an implicit Runge-Kutta scheme. We also construct in a systematic way a high-order quadrature which is optimal in terms of the number of points, which enables the use of mass lumping, up to P5 elements. We compare computational time and effort for several codes which are of high order in time and space and study their respective properties.