• Abstract

This paper proposes a high-order maximum-principle-preserving (MPP) conservative scheme for convection-dominated diffusion equations. For high-order spatial discretization, we first use the fifth-order weighted compact nonlinear scheme (WCNS5) for the convection term and the sixth-order central difference scheme for the diffusion term. Owing to the nonphysical oscillations caused by the high-order scheme, we further adopt a parameterized MPP flux limiter by modifying a high-order numerical flux toward a lower-order monotone numerical flux to achieve the maximum principle. Subsequently, the resulting spatial scheme is combined with third-order strong-stability-preserving Runge-Kutta (SSPRK) temporal discretization to solve convection-dominated diffusion problems. Several one-dimension (1D) and two-dimension (2D) numerical experiments show that the proposed scheme maintains up to fifth-order accuracy and strictly preserves the maximum principle. The results indicate the proposed scheme’s strong potential for solving convection-dominated diffusion and incompressible flow problems.

• General Summary

• Abstract

A simple weighted essentially non-oscillatory (SWENO) scheme for solving the convection-diffusion equation is proposed in this paper, where a seventh-order SWENO method and the third-order strong stability preserving (SSP) Runge-Kutta method are adopted for discretizing the space and time, respectively. Then the hyperbolic and diffusive part can achieve the seventh- and sixth-order accuracy, respectively. The proposed method has the following advantages. Firstly, negative linear weights are avoided. Secondly, one reconstruction with one stencil is required for the computation of convective and diffusive fluxes. Finally, the new method does not require the transformation while the diffusion coefficients are degenerate. Numerical examples demonstrate that the new method can achieve sixth-order accuracy in the smooth region and guarantee non-oscillatory properties for the discontinuous problems for one- and two-dimensional cases.

• General Summary

• Abstract

A class of efficient multistep methods for forward backward stochastic differential equations (FBSDEs) are proposed and studied in this paper. According to the characteristic of our multistep methods and replacing the corresponding Brownian motion increments with appropriate two-point distributed random variables, we obtain a very efficient algorithm to approximate the conditional expectations involved in the multistep methods. It is thanks to this efficient algorithm that we can obtain a class of efficient fully discrete form of high-order methods for the FBSDEs. Finally, some numerical results are presented to illustrate the efficiency of our multistep methods.

• General Summary