Large Time Step Explicit Schemes for Partial Differential Evolution Equations

Abstract

We consider the large time step (LTS) method for hyperbolic conservation laws, originally proposed by LeVeque in a series of papers over thirty years ago. In particular we have designed a local multi-point LTS scheme, denoted LTS Constant-Diffusion-$\hat{k}$ (LTS CD$\hat{k}$) scheme, which possesses an inherent natural mechanism for adding numerical viscosity. With this scheme we observe non-oscillating solutions, where other LTS schemes tend to create spurious oscillations for high Courant numbers. The scheme is first order accurate and total variation diminishing (TVD). Its robustness is evaluated by performing simulations for the Euler equations. With the LTS CD$\hat{k}$ scheme we successfully simulate one of the test cases presented in [\textit{LeVeque, R. J. (1985). A large time step generalization of Godunovs method for systems of conservation laws. SIAM Journal on Numerical Analysis, 22(6):1051 1073.}], which gave poor results with the LTS Godunov scheme. The oscillations observed by LeVeque for high Courant numbers are smeared with our diffusive LTS CD$\hat{k}$ scheme. For all problems considered in this thesis, the LTS CD$\hat{k}$ scheme yields solutions without oscillations. Our LTS CD$\hat{k}$ scheme hence provides a significant improvement in robustness compared to previously studied LTS schemes, and is a main result of this thesis.

We give a recipe for constructing higher order LTS schemes, and analyze convergence for the LTS schemes up to third order applied to the linear advection equation. A second order LTS CD$\hat{k}$ scheme is tested for the Sod shock tube problem, which gives very accurate results, but with some oscillations around discontinuities. These higher order schemes are not TVD. We have performed a von Neumann stability analysis to evaluate if they are linearly stable.

Finally we extend the LTS method to the linear constant coefficient convection-diffusion equation, which is a parabolic partial differential evolution equation, by matching the physical viscosity with the numerical viscosity in the modified equation for the LTS CD$\hat{k}$ scheme. Also for this equation we propose a method for constructing higher order schemes. A convergence analysis is performed for a second order LTS CD$\hat{k}$ scheme for different ratios of convection to diffusion, verifying the expected second order convergence.