Implicit Scheme for Hyperbolic Conservation Laws Using Nonoscillatory Reconstruction in Space and Time

The efficiency of high order accurate schemes for the solution of unsteady hyperbolic conservation laws is adversely affected by time-step restrictions that arise from monotonicity requirements. When applied to the solution of problems involving discontinuities, these restrictions render conventional high order implicit time integration schemes impractical. In the present study, a new single step second order implicit scheme that uses nonoscillatory reconstruction in space and time is presented. Both the spatial and temporal limiters are dependent on the evolving solution, and this nonlinearity allows for a circumvention of total variation diminishing bounds. Numerical results on model scalar and vector hyperbolic equations suggest that the scheme holds promise in achieving accurate and unconditionally nonoscillatory solutions.