Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem (Communications in Computational Physics, 2021)
We present a new conservative semi-Lagrangian finite difference weightedessentially non-oscillatory scheme with adaptive order. This is an extension of theconservative semi-Lagrangian (SL) finite difference WENO scheme in [Qiu and Shu,JCP, 230 (4) (2011), pp. 863-889], in which linear weights in SL WENO frameworkwere shown not to exist for variable coefficient problems. Hence, the order of accuracy is not optimal from reconstruction stencils. In this paper, we incorporate a recentWENO adaptive order (AO) technique [Balsara et al., JCP, 326 (2016), pp. 780-804]to the SL WENO framework. The new scheme can achieve an optimal high order ofaccuracy, while maintaining the properties of mass conservation and non-oscillatorycapture of solutions from the original SL WENO. The positivity-preserving limiter isfurther applied to ensure the positivity of solutions. Finally, the scheme is applied tohigh dimensional problems by a fourth-order dimensional splitting. We demonstratethe effectiveness of the new scheme by extensive numerical tests on linear advectionequations, the Vlasov-Poisson system, the guiding center Vlasov model as well as theincompressible Euler equations.
Keywords: Semi-Lagrangian; weighted essentially non-oscillatory; WENO adaptive order reconstruction; finite difference; mass conservation; Vlasov-Poisson; incompressible Euler.