To Study Large Time Step High Resolution Low Dissipative Schemes for Hyperbolic Conservation Laws

Abstract

Total Variation Diminishing (TVD) schemes are low dissipative and high resolution schemes but bounded by stability criterion CFL<1 for explicit formulation. Stability criteria for explicit formulation limits time stepping and thus increase computational cost (computational time, machine cost). Research in the field of large time step (LTS) scheme is an active field for last three decades. In present work, Zhan Sen Qian’s modified form of Harten LTS TVD scheme is studied and used to solve one dimensional benchmark test cases. SOD and LAX cases of shock tube problem are solved to understand the behavior of modified large time step scheme in regions of discontinuities and strong shock waves. The numerical results are found to be in good agreement with analytical results, except slight oscillations near contact discontinuity for larger values of K. Results also reveal that the discrepancy between numerical and analytical results near expansion fan, contact discontinuity and shock grows for larger values of K. Increase in discrepancy is due to the increase in truncation error. Truncation error strongly depends on step size and step size increases as CFL (or K) increases. In present work, the correction into the numerical formulation of characteristic transformation is discussed and the inverse characteristic transformations are performed using local right eigen vector in each cell interface location. This idea of extending Harten’s large time step method for hyperbolic conservation laws proved to be very useful as the results shows that the modified scheme is a high resolution low dissipative and efficient scheme for 1D test cases.

Keywords

• TVD scheme; Shock tube problem; Explicit scheme; Efficient scheme; 1D Euler equation.