Prof. Wai Sun Don, Xiao Wen, Yinghua Wang, Baoshan Wang, Xianjun Meng attended the 18th National Symposium on Numerical Methods in Fluids during Aug. 12-15, 2017.

Talk of Prof. Wai Sun Don

Title

Hybrid Compact-WENO Finite Difference Scheme for Hyperbolic Conservation Laws

Abstract

I will discuss some recent development of Bound-Preserving hybridization of high order nonlinear weighted essentially non-oscillatory (WENO) finite difference scheme and classical linear (finite difference and compact) scheme and non-classical (Fourier continuation) scheme (Hybrid), together with several high order shock sensors (multi-resolution, conjugate Fourier, Radial Basis function and Quantiles) to determine the smoothness of a solution of the nonlinear hyperbolic conservation laws. The advantages and disadvantages of the various components of the Hybrid schemes in terms of accuracy and efficiency as well as several relevant critical issues of the Hybrid scheme will also be discussed and illustrated with examples. Examples, including the one dimensional shock-entropy wave interaction and two dimensional classical Riemann IVP problems, March 10 double Mach reflection problems and detonation wave problems, regarding the efficiency and accuracy of the Hybrid scheme will be discussed.

Talk of Yinghua Wang

Title

Hybrid Compact-WENO Finite Difference Scheme for Hyperbolic Conservation Laws

Abstract

For discontinuous solutions of hyperbolic conservation laws, a hybrid scheme, based on the high order nonlinear characteristic-wise weighted essentially non-oscillatory (WENO) conservative finite difference scheme and the spectral-like linear compact finite difference scheme, is developed for capturing shocks and strong gradients accurately and resolving fine scale structures efficiently. The key issue in any hybrid scheme is the design of an accurate, robust, and efficient high order shock detection algorithm which is capable of determining the smoothness of the solution at any given grid point and time. An iterative adaptive multi-quadric radial basis function (IAMQ-RBF) method [Jung et al., Appl. Numer. Math. 61 (2011)] has been successfully used in the edges detection of the piecewise smooth functions. To reduce the ill-conditioning of the matrix system as well as to improve the efficiency of the IAMQ-RBF based shock detection algorithm, a domain decomposition technique, a novel fast Toeplitz matrix solver, and an outliers-detection algorithm are employed. In this study, we will investigate its applicability as the shock detector in the hybrid scheme and discuss its accuracy, efficiency, and other implementation issues in detail. Numerous classical one- and two-dimensional examples in shocked ow demonstrate that the developed hybrid scheme can reach a speedup of the CPU times by a factor up to 2-3 compared with the pure fifth order WENO-Z scheme.

Talk of Baoshan Wang

Title

Fast Iterative Adaptive Multi-Quadric Radial Basis Function Method for Edges Detection of Piecewise Functions

Abstract

In [J.H. Jung et al., Appl. Numer. Math. 61 (2011)], an iterative adaptive multi-quadric radial basis function (IAMQ-RBF) method has been developed for edges detection of the piecewise analytical functions. For a uniformly spaced mesh, the perturbed Toeplitz matrices, which are modified by those columns where the shape parameters are reset to zero due to the appearance of edges at the corresponding locations, are created. Its inverse must be recomputed at each iterative step, which incurs a heavy O(n^3) computational cost. To overcome this issue of efficiency, we develop a fast direct solver (IAMQ-RBF-Fast) to reformulate the perturbed Toeplitz system into two Toeplitz systems and a small linear system via the Sherman-Morrison-Woodbury formula. The O(n^2) Levinson-Durbin recursive algorithm that employed Yule-Walker algorithm is used to find the inverse of the Toeplitz matrix fast. Extensive and representative examples have demonstrated that the IAMQ-RBF-Fast based edges detection algorithm detects the edges in several classical test images not only accurately but also three times faster. Preliminary results in the density solution of the 1D Mach 3 extended shock-density wave interaction problem solved by the hybrid compact-WENO finite difference scheme with the IAMQ-RBF-Fast based shocks detection algorithm demonstrating an excellent performance in term of speed and accuracy, are also shown.