Development of a New Class of High order Implicit Non-Oscillatory Schemes for Conservation Laws

Abstract

Hyperbolic conservation laws allow for discontinuities to develop in the solution. In order to obtain non-oscillatory solutions near discontinuities and high gradient regions, numerical schemes have to satisfy conditions additional to linear stability requirements. These restrictions render high order implicit time integration schemes impractical since the allowable time-step sizes are not much higher than that for explicit schemes. In this work, an investigation is made on the factors that cause these time-step restrictions and two novel schemes are developed in an attempt to overcome the severity of these restrictions. In the first method, the order of accuracy of the {\em time integration} is lowered in high gradient and discontinuous regions. In the second method, the solution is reconstructed in {\em space and time} in a non-oscillatory manner. These concepts are evaluated on model scalar and vector hyperbolic conservation equations. The ultimate objective of this work is to develop a scheme that is accurate and unconditionally non-oscillatory.