Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Composite overlapping meshes for the solution of partial differential equations
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
A note on the effect of artificial viscosity on solutions of conservation
Applied Numerical Mathematics
Flux-corrected transport I. SHASTA, a fluid transport algorithm that works
Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
High resolution schemes for hyperbolic conservation laws
Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
On some numerical dissipation schemes
Journal of Computational Physics
An integral evolution formula for the wave equation
Journal of Computational Physics
Difference Approximations for the Second Order Wave Equation
SIAM Journal on Numerical Analysis
Strongly consistent marching schemes for the wave equation
Journal of Computational Physics
High order marching schemes for the wave equation in complex geometry
Journal of Computational Physics
Difference Approximations of the Neumann Problem for the Second Order Wave Equation
SIAM Journal on Numerical Analysis
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
A High-Order Accurate Parallel Solver for Maxwell’s Equations on Overlapping Grids
SIAM Journal on Scientific Computing
Finite volume transport schemes
Numerische Mathematik
On sub-linear convergence for linearly degenerate waves in capturing schemes
Journal of Computational Physics
Uniform Asymptotic Stability of Strang's Explicit Compact Schemes for Linear Advection
SIAM Journal on Numerical Analysis
Numerical methods for solid mechanics on overlapping grids: Linear elasticity
Journal of Computational Physics
Grid stabilization of high-order one-sided differencing II: Second-order wave equations
Journal of Computational Physics
Richardson Extrapolation for Linearly Degenerate Discontinuities
Journal of Scientific Computing
Hi-index | 31.45 |
We develop new high-order accurate upwind schemes for the wave equation in second-order form. These schemes are developed directly for the equations in second-order form, as opposed to transforming the equations to a first-order hyperbolic system. The schemes are based on the solution to a local Riemann-type problem that uses d'Alembert's exact solution. We construct conservative finite difference approximations, although finite volume approximations are also possible. High-order accuracy is obtained using a space-time procedure which requires only two discrete time levels. The advantages of our approach include efficiency in both memory and speed together with accuracy and robustness. The stability and accuracy of the approximations in one and two space dimensions are studied through normal-mode analysis. The form of the dissipation and dispersion introduced by the schemes is elucidated from the modified equations. Upwind schemes are implemented and verified in one dimension for approximations up to sixth-order accuracy, and in two dimensions for approximations up to fourth-order accuracy. Numerical computations demonstrate the attractive properties of the approach for solutions with varying degrees of smoothness.