A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws

doi:10.1016/j.jcp.2016.06.027

Journal of Computational Physics

Volume 322, 1 October 2016, Pages 559-585

https://doi.org/10.1016/j.jcp.2016.06.027 Get rights and content

Abstract

For a nonlinear scalar conservation law in one-space dimension, we develop a locally conservative semi-Lagrangian finite difference scheme based on weighted essentially non-oscillatory reconstructions (SL-WENO). This scheme has the advantages of both WENO and semi-Lagrangian schemes. It is a locally mass conservative finite difference scheme, it is formally high-order accurate in space, it has small time truncation error, and it is essentially non-oscillatory. The scheme is nearly free of a CFL time step stability restriction for linear problems, and it has a relaxed CFL condition for nonlinear problems. The scheme can be considered as an extension of the SL-WENO scheme of Qiu and Shu (2011) [2] developed for linear problems. The new scheme is based on a standard sliding average formulation with the flux function defined using WENO reconstructions of (semi-Lagrangian) characteristic tracings of grid points. To handle nonlinear problems, we use an approximate, locally frozen trace velocity and a flux correction step. A special two-stage WENO reconstruction procedure is developed that is biased to the upstream direction. A Strang splitting algorithm is used for higher-dimensional problems. Numerical results are provided to illustrate the performance of the scheme and verify its formal accuracy. Included are applications to the Vlasov–Poisson and guiding-center models of plasma flow.

Introduction

We develop a semi-Lagrangian finite difference WENO (SL-WENO) scheme to approximate the scalar conservation law $u_{t} + {(f (u))}_{x} = 0, x \in R, t > 0,$ $u (x, 0) = u^{0} (x), x \in R,$ with the possibly nonlinear flux $f (u) = f (u; x, t)$ . Moreover, multiple space dimensions will be treated using operator splitting.

The semi-Lagrangian computational scheme uses a mixture of Eulerian and Lagrangian reference frames in the sense that it has a fixed (Eulerian) numerical grid but advances each time step by evolving the partial differential equations by propagating information along (Lagrangian) characteristic curves. A semi-Lagrangian scheme does not inherently suffer the CFL time step restriction of an Eulerian approach. In consequence, longer time step evolution can be achieved, potentially resulting in less computational effort and less numerical diffusion.

In 2011, Qiu and Shu [1] (see also [2]) developed a semi-Lagrangian finite difference WENO method for linear scalar conservation laws for which $f (u) = a u$ for some advection velocity $a (x, t)$ . In fact, they could develop the linear weights needed in the WENO reconstruction [3], [4] only when the velocity a is constant. For variable advection velocity, they developed weights based on heuristic empirical criteria.

In this paper we develop a locally mass conservative SL-WENO scheme for nonlinear scalar conservation laws. In the linear case, it is an extension of the scheme of Qiu and Shu [1]. It reduces to their scheme if the advection velocity a is constant and linear WENO weights are used; however, our scheme has exact WENO linear weights in the case of variable advection problems. Our scheme is new in the case of nonlinear problems, and it can be viewed as an extension of related work on finite volume Eulerian–Lagrangian WENO schemes [5], [6], in which characteristic tracing is incorporated only in an approximate way. For simplicity of exposition, we will describe only the third order SL-WENO scheme (SL-WENO3), but in principle any order scheme can be implemented.

Our finite difference scheme is based on the standard sliding average formulation of Shu and Osher [7], [8], but the flux function is defined using WENO reconstructions of (semi-Lagrangian) characteristic tracings of grid points [1]. To handle nonlinear problems for which the trace velocity in unknown, we use a fixed, approximate trace velocity (as is done in, e.g., [9], [6] and in arbitrary Lagrangian–Eulerian (ALE) schemes [9]). This approximate trace velocity is locally frozen to avoid numerical difficulties [6]. The introduction of an approximate trace velocity means that we do not trace the exact characteristic curves, and necessitates the use of a flux correction step [6], which is implemented using the approach of Levy, Puppo, and Russo [10] developed for CWENO schemes.

Unlike the direct WENO reconstructions used by Qiu and Shu [1], we develop a two-stage WENO reconstruction procedure. We first reconstruct cell averages of the numerical flux function, and then we reconstruct the flux function at the needed point. In our procedure, the linear WENO weights always exist. We nonlinearly weight at each stage using a smoothness indicator that is based on the underlying smoothness of the solution u; that is, rather than using the smoothness of the cell average fluxes in the second stage, we use the smoothness indicator for u in both stages.

For nonlinear problems, the scheme requires a special upstream bias in the computation of the flux function between two grid points, since nonsmooth shocks may develop. The Rankine–Hugoniot shock speed is computed assuming a shock based on the solution u between the two grid points (i.e., the Roe speed [11]). The direction of this assumed shock determines the wave direction and consequent upstream biasing in our scheme.

The flux correction step requires a relaxed CFL constraint, given later in (42) [12], [6]. The constraint is based on the speed of the difference between the true value of $\partial f (u) / d u$ and the fixed approximate trace velocity. If these are approximately equal, the constraint is very mild.

For multi-dimensional problems, we use a standard, usually second order Strang splitting which decouples the differential equation into a sequence of one-dimensional problems (see (53)). The advantage of a finite difference scheme is that the splitting error manifests itself as temporal error, because the splitting does not induce a shearing of the variable coefficients, and so the scheme maintains its formal high order spatial accuracy [2].

We present a detailed description of our formally third order SL-WENO3 scheme in the next section. Sections 3 and 4 are devoted to numerical results of SL-WENO3 and SL-WENO5 in one and two space dimensions, respectively. In Section 5 we apply our new scheme to two models of plasma transport: (1) the Vlasov–Poisson system, which has a constant advection within the Strang split scheme, and (2) the guiding-center model [13], [14], which has a nonconstant advection. We conclude briefly in the last section.

Section snippets

The semi-Lagrangian finite difference WENO scheme

The development of our semi-Lagrangian finite difference WENO (SL-WENO) scheme for nonlinear problems parallels and extends the development of the finite difference scheme given by Qiu and Shu [1] for linear problems. For simplicity of exposition, we describe the third order SL-WENO scheme.

Partition time and space as $t^{0} < t^{1} < t^{2} < \dots$ and $\dots < x_{- 1} < x_{0} < x_{1} < \dots$ , respectively, where $Δ x = x_{i} - x_{i - 1}$ is constant. We approximate $u (x_{i}, t^{n})$ as $u_{i}^{n}$ . Integrate (1) over $[t^{n}, t^{n + 1}]$ and evaluate at a grid point $x_{i}$ to obtain $u_{i}^{n + 1}$

Some numerical results in one space dimension

We present several examples of our numerical scheme to test its accuracy and performance. Some of the examples are for the linear equation $u_{t} + {(a (x, t) u)}_{x} = 0,$ for which we specify only the velocity $a (x, t)$ . In that case, unless stated otherwise, we use the simple trace velocity $v_{i + 1 / 2} = a (x_{i}, t^{n})$ .

Some numerical results in two space dimensions

There are standard ways to apply our finite difference scheme to two-space dimensional problems. For $u_{t} + {(f (u))}_{x} + {(g (u))}_{y} = 0, x, y \in R, t > 0,$ we might simply compute the analogue of (7), which involves a sliding average function in the y-variable $K$ . The conservative scheme is based on $u_{i j}^{n + 1} = u_{i j}^{n} - \frac{1}{Δ x} [H (x_{i + 1 / 2, j}) - H (x_{i - 1 / 2, j})] - \frac{1}{Δ y} [K (x_{i, j + 1 / 2}) - K (x_{i, j - 1 / 2})] .$ However, we prefer to view the treatment of space as a Strang splitting. For a first order Strang splitting, we compute the two-stage scheme ${\tilde{u}}_{i j} = u_{i j}^{n} - \frac{1}{Δ x} [$

The Vlasov–Poisson (VP) system in one space dimension

The well-known nondimensionalized Vlasov–Poisson (VP) system in one space dimension is $f_{t} + v f_{x} + E (t, x) f_{v} = 0,$ $E (t, x) = - ϕ_{x}, - ϕ_{x x} (t, x) = ρ (t, x),$ where x and v are the coordinates in the phase space $(x, v) \in R \times R$ , E is the electric field, ϕ is the self-consistent electrostatic potential, and $f (t, x, v)$ is the probability distribution function describing the probability of finding a particle with velocity v at position x at time t. The probability distribution function couples to long ranges fields via the

Conclusions

We developed a new semi-Lagrangian finite difference WENO scheme for nonlinear scalar conservation laws in one or more space dimensions. The scheme is locally mass conservative, formally high-order accurate in space, has small time truncation error, and is essentially non-oscillatory. It is subject to a relaxed CFL time step stability condition (42). The scheme can be considered as an extension of the SL-WENO scheme of Qiu and Shu [1] for linear problems, and as a finite difference version of

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¹: Supported in part by Taiwan National Science Council grant NSC 102-2115-M-110-010-MY3.

²: Supported in part by U.S.National Science Foundation grant DMS-1418752.

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A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws

Abstract

Introduction

Section snippets

The semi-Lagrangian finite difference WENO scheme

Some numerical results in one space dimension

Some numerical results in two space dimensions

The Vlasov–Poisson (VP) system in one space dimension

Conclusions

Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow

J. Comput. Phys.

Conservative semi-Lagrangian finite difference WENO formulations with applications to the Vlasov equation

Commun. Comput. Phys.

Weighted essentially non-oscillatory schemes

J. Comput. Phys.

Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws

An Eulerian–Lagrangian WENO finite volume scheme for advection problems

J. Comput. Phys.

Efficient implementation of essentially non-oscillatory shock capturing schemes

J. Comput. Phys.

Efficient implementation of weighted ENO schemes

J. Comput. Phys.

Arbitrary Lagrangian–Eulerian methods

Central WENO schemes for hyperbolic systems of conservation laws

Math. Model. Numer. Anal.

Approximate Riemann solvers, parameter vectors, and difference schemes

J. Comput. Phys.

A moving mesh method for one-dimensional hyperbolic conservation laws

SIAM J. Sci. Comput.