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.

[1]  C. Angelopoulos.  High resolution schemes for hyperbolic conservation laws , 1992 .

[2]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[3]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[4]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[5]  H. Huynh,et al.  Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping , 1997 .

[6]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[7]  T. Pulliam.  Time accuracy and the use of implicit methods. [in CFD , 1993 .

[8]  S. Osher,et al.  Uniformly high order accuracy essentially non-oscillatory schemes III , 1987 .

[9]  R. LeVeque.  Approximate Riemann Solvers , 1992 .

[10]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[11]  A. Suresh,et al.  The Reconstruction Problem Revisited , 1999 .

[12]  Shaun A. Forth,et al.  A Second Order Accurate, Space-Time Limited, BDF Scheme for the Linear Advection Equation , 2001 .

[13]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[14]  S. Osher,et al.  Uniformly High-Order Accurate Nonoscillatory Schemes. I , 1987 .

[15]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[16]  James D. Baeder,et al.  Concepts and Application of Time-Limiters to High Resolution Schemes , 2003, J. Sci. Comput..