Hybrid Compact-WENO Schemes For Detonation Waves Simulations

In this thesis, hybrid compact-WENO schemes (Hybrid) and their applications for detonation wave are studied.The explosion is a very rapid energy release process of physical or chemical, it is the complex process with a reaction layer produced in the forefront of shock. Despite the explosion has been studied for many years, because of the importance of practice, explosion theory and numerical simulation studies are still popular in the field of research. Direct numerical simulations of fine scale and delicate structures of detonation waves demand the use of highly accurate and efficient numerical schemes, which must be able to resolve a very broad range of length and time scales. In recent years, WENO (Weighted Essentially Non-Oscillation) schemes, previously used to solve the equations of hyperbolic conservation laws contain solutions of the shock and the small-scale structure, and are used to simulate the blast wave to achieve good results. The use of a dynamic set of substencils where a nonlinear convex combination of lower order polynomials adapts either to a higher order polynomial approximation at smooth parts of the solution, or to a lower order polynomial approximation that avoids interpolation across discontinuities, yields a local rate of convergence that goes from order r at the non-smooth parts of the solution, to order (2r-1)when the convex combination of local lower polynomials is applied at smooth parts of the solution. However, WENO schemes computationally intensive, and the explosion shock wave itself, centralized distribution, so if you can handle the explosion segments:shock at the use WENO schemes, smooth at the use of linear schemes, that is a very natural idea to solve pure WENO schemes computationally intensive problems. Hybrid central-WENO scheme for detonation wave simulation has good simulation results. Hybrid compact-WENO schemes is a good combination, this hybrid schemes has been used in other shock problems, but it has not been applied to the detonation waves.Compact difference scheme (Compact) is an implicit scheme, and compared to traditional finite difference scheme, which has higher accuracy, better stability, better resolution (for high-frequency waves), while fewer boundaries points need to be handle. It is systematically investigated that the performance of a hybrid compact finite difference scheme and characteristic-wise weighted essentially non-oscillatory (WENO) finite difference scheme (Hybrid) for the detonation waves simulations on a uniformly discretized Cartesian domain. The Hybrid scheme is used to keep the solutions parts displaying high gradients and discontinuities always captured by the WENO-Z scheme in an essentially non-oscillatory manner while the smooth parts are highly resolved by an efficient and accurate Compact scheme and to speedup the computation of the overall scheme. A high order multi-resolution analysis by A. Harten is used to capture the location of discontinuities to maintain the high order/resolution of the Hybrid scheme.In this study, the 3th-,5th-,7th-order WENO-Z schemes and one spectral-like higher order Compact scheme are conjugated in the discontinuous and smooth parts, respectively. Numerical oscillations generated by the Compact scheme are mitigated by the high order filtering. The high order multi-resolution algorithm is employed to detect the smoothness of the solution. The numerical comparisons between the WENO-Z scheme and the Hybrid scheme in the simulations of one-dimensional shock-entropy wave interaction show that the Hybrid scheme conjugated with the 5th-order WENO-Z scheme and the 6th-order Compact scheme is the best than the other combinations if the balance of accuracy and efficiency is considered and the Hybrid scheme is more efficient, less dispersive and less dissipative than the WENO-Z scheme. The paper apply the 5th-order WENO-Z scheme and corresponding Hybrid scheme (conjugated with the 5th-order WENO-Z scheme and the 6th-order Compact scheme) in simulating one-dimensional Riemann initial value problems (123, Sod and Lax), two-dimensional double mach reflection problem, classical stable detonation waves two-dimensional detonation waves and two-dimensional shock diffraction problem, those numerical results show that the Hybrid scheme can speedup the computation to two or three times faster than the WENO-Z scheme under the same reaction zone resolution without reducing the accuracy, and the MR method captures the location of the detonation front very accurately.