Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
ICOSAHOM '89 Proceedings of the conference on Spectral and high order methods for partial differential equations
Flux-corrected transport I. SHASTA, a fluid transport algorithm that works
Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
Formulations of artificial viscosity for multi-dimensional shock wave computations
Journal of Computational Physics
The construction of compatible hydrodynamics algorithms utilizing conservation of total energy
Journal of Computational Physics
An adaptive numerical scheme for high-speed reactive flow on overlapping grids
Journal of Computational Physics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
A high-resolution Godunov method for compressible multi-material flow on overlapping grids
Journal of Computational Physics
An evaluation of the FCT method for high-speed flows on structured overlapping grids
Journal of Computational Physics
Journal of Computational Physics
Deforming composite grids for solving fluid structure problems
Journal of Computational Physics
Upwind schemes for the wave equation in second-order form
Journal of Computational Physics
Numerical methods for solid mechanics on overlapping grids: Linear elasticity
Journal of Computational Physics
Richardson Extrapolation for Linearly Degenerate Discontinuities
Journal of Scientific Computing
Hi-index | 31.47 |
A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t^1^/^2, yielding a convergence rate of 1/2. Self-similar heuristics for higher-order discretizations produce a growth rate for the layer thickness of @Dt^1^/^(^p^+^1^) which yields an estimate for the convergence rate as p/(p+1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations.