Inverse Lax–Wendroff procedure for numerical boundary conditions of convection–diffusion equations

Abstract

We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure [28], in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, have been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one-and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.

Essentially non-oscillatory and weighted essentially non-oscillatory schemes

“…We refer to Goldberg andTadmor (1978, 1981) for earlier discussions on this procedure. This procedure has been generalized to moving boundaries in Tan and Shu (2011), and to convection-diffusion equations in Lu et al (2016). A simplified version, for which the (more expensive) ILW procedure is only applied to several lower-order derivatives and standard extrapolation is applied to the remaining ones, has been introduced in Tan, Wang, Shu and Ning (2012) and analysed for stability in Zhang (2016, 2017).…”

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

“…So far, the ILW method has been mostly used on hyperbolic equations including conservation laws [216,217,218,146] and Boltzmann type models [71], which involve only first order spatial derivatives in the equations. Very recently, Lu et al extended this methodology to convection-diffusion equations [157]. It turns out that this extension is highly non-trivial, as totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes.…”

“…It turns out that this extension is highly non-trivial, as totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. A careful combination of these two boundary treatments has been designed in [157], in order to obtain a stable and accurate boundary condition for high order finite difference schemes when applied to convection-diffusion equations, regardless of whether they are convection or diffusion dominant.…”

“…The ILW and SILW methods have enabled simple finite difference schemes on Cartesian meshes to be used for solving complex geometry problems with shocked solutions [216,217,219,157]. These methods hold a good potential in wider applications.…”

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

“…This approach is of course delicate, especially with finite-difference discretizations on non-conforming meshes. In this context, a very successful technique is the inverse Lax-Wendroff approach, which was introduced in [40], rendered more computationally efficient in [41], and further studied and extended for example in [24,27,28]; a quite up-to-date review may be found in [39]. A modified procedure enhancing its accuracy and stability has been proposed in [43].…”