Shihao Liu

My research area is numerical method of partial differential equations (PDEs). In particular, I focus on the high order accuracy boundary treatment based on finite difference methods with non body-fitted mesh. More specifically, I mainly study a method called "inverse Lax-Wendroff" (ILW) method, which is first proposed by Tan and Shu (in J. Comput. Phys. 229: 8144-8166, 2010). In my research, I made a series of promotion and improvement to this algorithm for different situations. Both of stationary and moving boundary problems were under consideration. The improved algorithm was utilized in different types of equations, especially the hyperbolic conservation laws, in general convection dominated problems, as well as the convection-diffusion equations. The solutions of these equations may contain discontinuties or very sharp interfaces near the boundary. Therefore, my main research is to design a stable, accurate and efficient numerical boundary scheme, that has high order accuracy for smooth solutions and non-oscillatory property for shock solution near the boundary. Some of my work is briefly described in the following.

1.A new type of simplified inverse Lax-Wendroff framework

The original ILW method requires heavy computation to approximate high order space derivatives. Several popular ILW methods simplify the original method, but they still require a lot of computation, especially for high-dimensional system. We proposed a new type of ILW framework for some different kinds of partial differential equations. This new ILW framework decomposes the construction of ghost points into two steps: interpolation and extrapolation instead of the Taylor expansion framework of the previous method. Under the premise of high order accuracy and stability, this new proposed method would reduce the number of high order derivatives needed at the boundary. And therefore, it can avoid the complicated computation of high order ILW process and improve the computational efficiency. The linear stability of the new ILW method was given by using the eigenvalue analysis method, demonstrating the stability of the new method. This framework was first proposed for the initial boundary value problems of hyperbolic conservation laws, and then extended to the initial boundary value problems of diffusion equations and convective-diffusion equations. Numerical experiments showed that the new ILW method is stable, efficient, and high order accuracy. Our future work is to extend this new ILW framework so it can be utilized in other situations, such as different types of equations, for example, the equations of elasticity, different types of boundary condition (Robin condition), and moving boundary problem.

2.Moving boundary inverse Lax-Wendroff method for viscous flows

Applying ILW idea to construct a high order stable numerical treatment for moving boundary is one of the key problems that scholars are concerned about. We proposed a new type of ILW method for the moving boundary problem of the convection-diffusion equations. Firstly, the diffusion-dominated equations and the convection-dominated ones were considered, and the corresponding boundary treatment methods were designed respectively. A convex combination of those two different boundary treatments was given for the general convection-diffusion problem. In particular, the weights of convex combination were carefully designed, resulting in a unified algorithm for pure convection, convection-dominated, convection-diffusion, diffusion-dominated and pure diffusion cases. Meanwhile, we used the third-order TVD Runge-Kutta method for time discretization.  In order to avoid the accuracy loss of time discretization and improve the computational efficiency, we designed a new approximation for the mixed derivatives on the boundary when constructing the ghost points at two intermediate stages of RK method. Finally, the proposed method was used to simulate the motion of rigid body under 2D and 3D shock waves, and the numerical results verified the feasibility and effectiveness of our method

3.Inverse Lax-Wendroff method with geometric boundary represented by signed distance function

Previous studies on ILW method did not involve the geometric representation of the boundary, and the geometric structure of computational domain of the numerical tests taken in previous literatures was relatively simple. However, if ILW method is to be used to solve problems emerging in engineering that involve complex geometric boundaries, it is inevitable to find a geometric boundary representation method suitable for ILW method. Signed distance function (SDF) representation is an implicit representation of geometric boundary that describes the signed distance from a point to a boundary. By comparing with other representation methods, we indicated the SDF representation is a relatively suitable geometric representation for ILW method, which can easily obtain the geometric information required in the ILW procedure. Based on SDF representation, we established SDF-based ILW framework, which showed high efficiency and robustness through numerical tests.

In addition, in many practical industrial applications, only point clouds and normal information of some geometric models can be obtained.  Therefore, it is also a problem to reconstruct a continuous SDF representation of these geometric models quickly and accurately based on this information. 

4.A numerical study of supersonic flow around the wedge

The ILW method was also used to simulate the supersonic flow around the wedge, and the critical angle of the wedge which produces the detached shock wave is studied. The supersonic flow around the wedge will produce oblique shock waves. When the angle of the wedge is small, the shock wave is attached. As the angle of the wedge increases to a certain critical value, detached shock wave will be generated. Li (in J. Math. Phys. 48, 123102, 2007) deduced that the critical angle should fall in an interval through theoretical analysis. We applied the ILW method to the solid wall boundary treatment of wedge surface and perform numerical simulation. Numerical simulation results are consistent with the theoretical analysis result of Li.