Study On Time Advancement Method Of Partial Differential Equations Based On High-order Numerical Schemes

The high-order scheme can improve the accuracy of the simulation under the same mesh quality and obtain more accurate numerical solutions.We propose a high-order implicit compact WCNS scheme,and apply the high-order WCNS scheme and WENO scheme to the Shallow Water Equations.In order to make the scheme meet the harmony under still water conditions,special treatment for the source term must be made.Numerical simulations of the classic dam-breaking problem show that the fifth-order WENO scheme has large dissipation and more serious smoothing phenomenon,while the fourth-order WCNS scheme has stronger shock capture ability,higher resolution,good symmetry and harmony.When the high-order scheme is applied to numerical simulations,if explicit time discretization is used,there will be some problems of long calculation time and low efficiency,especially for rigid terms or with uneven coarse meshes.Therefore,in this paper,the one-dimensional convection-diffusion equations and the two-dimensional Euler equations are numerically simulated to study the effects of different time-discrete schemes on the efficiency and errors when the high-order scheme numerical simulation is performed,including:(1)For the one-dimensional convection-diffusion equations,if the CPR scheme is used for the spatial discretization,the numerical simulation shows that the discrete convection term of the CPR scheme reaches the third-order accuracy,and the discrete diffusion term reaches the fourth-order accuracy.On the other hand,to study the influence of linear equations iterative convergence rate on CPU time,the TVD-RK3 scheme and IMEX-RK3 scheme are also used for the time discretization,respectively.From the numerical simulation for the convection dominance and diffusion dominance of the convection-diffusion equations,it is concluded that the IMEX-RK3 scheme has higher computational efficiency and smaller errors than the TVD-RK3 scheme when solving the one-dimensional convection-diffusion equations.(2)For the two-dimensional Euler equations,the space discretization adopts the hybrid scheme of WCNS and CPR,and the time discretization adopts the Euler1 scheme and the dual time scheme,respectively.The influence of the time discretization scheme on the numerical simulation of the problem of uneven grids with coarse and fine meshes is studied.For the case that the virtual time step can only be set to satisfy the minimum grid stability condition,in this paper,the local time step is applied to the value of the virtual time step to accelerate the speed of reaching the pseudo-steady state.Numerical results show that,compared with the first-order explicit Euler scheme,the CPU time of the dual-time method is shorter,and the error is the same order of magnitude;compared with the dual time scheme in which the virtual time uses a constant time step,the dual time scheme has shorter CPU time,and the obtained numerical errors are basically the same.