Semi-implicit methods for advection equations with explicit forms of numerical solution

Abstract

We present a parametric family of semi-implicit second order accurate numerical methods for non-conservative and conservative advection equation for which the numerical solutions can be obtained in a fixed number of forward and backward alternating substitutions. The methods use a novel combination of implicit and explicit time discretizations for one-dimensional case and the Strang splitting method in several dimensional case. The methods are described for advection equations with a continuous variable velocity that can change its sign inside of computational domain. The methods are unconditionally stable in the non-conservative case for variable velocity and for variable numerical parameter. Several numerical experiments confirm the advantages of presented methods including an involvement of differential programming to find optimized values of the variable numerical parameter.

Semi-implicit high resolution numerical scheme for conservation laws

“…For a fixed value of ω i ≡ ω and l i ≡ 1 the method is second order accurate for smooth solutions if either f ≡ f + or f − ≡ f , see the Appendix. In the case of linear advection equation, the scheme is unconditionally stable for ω i ≥ 0 having no restriction on the choice of τ due to the stability, see the proof in [14].…”

“…Remark 1. To derive (14) we have supposed, among others, that u n+1 i−1 = u n i . As we show later, the case u n+1 i−1 = u n i can happen only if ω i−1 = 0, when the scheme (5) takes the simpler form,…”

“…Note that we keep the mixed derivative in (36), so we follow the Lax-Wendroff procedure of replacing every time derivative using the PDE ∂ t u = −∂ x f only partially [13,9,15,14]. Now applying the following approximations in ( 36)…”

“…In [15,14] it is shown that the schemes can be derived by considering and approximating mixed spatial-temporal derivatives in the Taylor expansion of the solution when using the Lax-Wendroff (LW) procedure only partially. The idea of not using LW to completely replace the time derivatives with the space derivatives using PDE is used also in different contexts in [13,9,10].…”

“…For this type of problems, the numerical scheme must be conservative to approximate correctly the movement of shock waves and the scheme must be nonlinear even for linear problems if higher resolution than the first order scheme with non-oscillatory numerical solutions is required. In this context, we introduce a novel semi-implicit method for some representative models of hyperbolic systems extending the approach for linear problems in [15,14] with some ideas from [13,25] for nonlinear problems.…”