Large Time Step Scheme Behaviour with Different Entropy Fix: Large Time Step Scheme Behaviour with Different Entropy Fix - Ihtram ul Haq; Hussain, Mukkarum; Iqbal, Muhammad Jawed; Siddiqui, Noor Fatima (Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences, 2021)
The progress of numerical techniques for scalar and one dimensional Euler equation has been a great interest of researchers in the field of CFD for decades. In 1986, Harten developed a high resolution and efficient large time step (LTS) explicit scheme for scalar problems. Computation of nonlinear wave equation depicts that Harten’s LTS scheme is a high resolution and efficient scheme. However, computations of hyperbolic conservation laws show some spurious oscillations in the vicinities of discontinuities for larger values of CFL. Zhan Sen Qian investigated this issue and suggested to perform the inverse characteristic transformations by using the local right eigenvector matrix at each cell interface location to overcome these spurious oscillations. Harten and Qian both used Roe’s approximate Riemann solver which has less artificial viscosity than exact method at sonic points. The reduced artificial viscosity reduces the accuracy of Roe's method at sonic points. Roe's approximate Riemann solver cannot capture the finite spread of expansion fans due to the inadequate artificial viscosity at expansive sonic points. As a consequence of this expansion shocks that are nonphysical may occur. The existence of the expansion shock is said to violate the entropy condition. A variety of entropy fix formulae for Roe scheme have been addressed in the literature. In present work large time step total variation diminishing (LTS TVD) scheme developed by Harten and improved by Qian have been tested with different entropy fix and its effect has been investigated. Computed results are analyzed for merits and shortcomings of different entropy fix with large time step schemes.