In data-driven modeling, Bayesian sparse identification method with Laplace priors was found and confirmed to recover sparse coefficients of governing partial differential equations(PDEs) by spatiotemporal data from measurement or simulation. Verification results of Bayesian sparse identification method for various canonical models (KdV equation, Burgers equation, Kuramoto-Sivashinsky equation, reaction-diffusion equations, nonlinear Schr dinger equation and Navier-Stokes equations) are compared with those of Rudy's PDE-FIND algorithm. Very well agreement between these two methods shows Bayesian sparse method has strong identification capability of PDE. However, it is also found that the Bayesian sparse method is much more sensitive to noise, which may identify more extra terms. In addition, relatively small error variances of Bayesian sparse solutions are obtained and exhibit clearly the successful identification of PDE.

We applie adaptive multi-resolution method to numerical simulation of stiffened-gas equation of state based on CJ model. A conservation sharp interface method of multiphase flow, in which the interface problem is tracked and solved by level-set method and ghost fluid method, is adopted. This method handles long physical time scale interactions across interface well and reduces conservative associated errors. In addition, pyramid data structure and adaptive multi-resolution analysis are used to improve storage efficiency and calculation efficiency of the algorithm. Finally, numerical examples are given to demonstrate stability and effectiveness of the adaptive multi-resolution method in numerical simulation of reactive multiphase flows.

We present a generalized Riemann problem based, remapping free moving mesh scheme to simulate reactive flows. The scheme is based on solution of a generalized Riemann problem on moving meshes and is explicitly remapping free. In construction of numerical fluxes, we use generalized Riemann problem scheme to get high accuracy. A variational approach is applied to generate an adaptive moving mesh to get high resolution in reactive zone. The scheme can not only keep the mesh quality,but also avoid efficiently numerical errors induced by an explicit remapping process in arbitrary Lagrangian-Eulerian methods. Numerical experiments, including Chapmann-Jouguet detonation and unstable detonation, demonstrate accuracy and robustness of the scheme. It shows that the generalized Riemann problem moving mesh method performs well for reactive flows with both discontinuities and smooth structures.

We give a semi-Lagrangian scheme for Vlasov-Poisson equation using third order upwind interpolation polynomial with limiter. The scheme is conservative and keeps solution positive. We use the scheme to compute typical examples that include Landau damping, two stream instability and symmetric two stream instability. These simulations are compared with other numerical results. In conclusion, the conservative scheme works well in solving Vlasov-Poisson equation.

We present an efficient method to simulate reactive flow for general equation of states in two dimensions. Two kinds of nonideal equation of states for compressible fluid model coupling with reactive rate equation are concerned. The important aspect is to deal with mixture of different phase in one cell, which inevitably happens in Euler method for reactive flows. Physical variables such as pressure,velocity and speed of sound in each cell are reconstructed which results in nonlinear algebra equations. They are used to obtain flux by HLLC Riemann solver. Numerical examples of detonation in two dimensions with different equation of states demonstrate accuracy and robustness of the method.

We give discretization for fluid system in integral form with arbitrary moving velocity on unstructured meshes, where re-mapping step in ALE method to interpolate flow variables between old mesh and new one can be avoided. We give three kinds of velocity for different part of computational domain, and apply the scheme to pitching Naca0012 airfoil with moving boundary. It shows that the scheme is efficient and accurate.

We concern extension of gas-kinetic scheme of BGK type to reactive fluids, and develop an adaptive moving meshes BGK scheme (AMMBGK). We derive systems from a mass fraction BGK model for detonation fluids, including both inviscid and viscous reactive flow systems. Then, based on a BGK type scheme and splitting method that splits system into fluid part and part of energy released from reaction process, we present a mass fraction BGK scheme on moving meshes for reactive flows. Numerical results validate availability of the gas-kinetic scheme in simulation of reactive fluids.

Based on WENO reconstruction,a high-order (at least third order) moving mesh kinetic scheme is presented for compressible flows.It employs the frame of remapping-free ALE-type kinetic method to discrete Euler equation.Adaptive moving mesh approach is used to move meshes.Kinetic flux evolution is used to construct numerical fluxes in order to achieve their time-accuracy.Numerical results demonstrate accuracy and robustness of the scheme.

Combining reference Jacobian method(RJM) with advancing-front method,we present a strategy named advancing reference Jacobian method(ARJM).It advances the optimization process step by step from one part of the computational region boundary to the remaining parts.In each step,two neighboring rows(or columns) are taken as the boundaries of the sub-region and the rear row(column) nodes are the optimized ones and the middle row(or column) is optimized by RJM. Analyses and numerical experiments show that the ARJM is much faster than RJM.The geometric qualities of rezoned grids by ARJM are equal to or even better than those by RJM.The rezoned grids obtained by ARJM are closer to Lagrangian grids than those by RJM.

The moving least squares particle hydrodynamics (MLSPH) is presented and a 1D MLSPH algorithm is discussed.To reduce oscillations near contact discontinuities,two meshfree Riemann problem initial value methods with different accuracy are introduced.The effect of contact interaction between particles is decreased by the Riemann solver.Efficiency of the method is demonstrated by numerical tests of a 1D shock tube.