Selective monotonicity preservation in scalar advection

Abstract

An efficient method for scalar advection is developed that selectively preserves monotonicity. Monotonicity preservation is applied only where the scalar field is likely to contain discontinuities as indicated by significant grid-cell-to-grid-cell variations in a smoothness measure conceptually similar to that used in weighted essentially non-oscillatory (WENO) methods. In smooth regions, the numerical diffusion associated with monotonicity-preserving methods is avoided. The resulting method, while not globally monotonicity preserving, allows the full accuracy of the underlying advection scheme to be achieved in smooth regions. The violations of monotonicity that do occur are generally very small, as seen in the tests presented here. Strict positivity preservation may be effectively and efficiently obtained through an additional flux correction step.

The underlying advection scheme used to test this methodology is a variant of the piecewise parabolic method (PPM) that may be applied to multi-dimensional problems using density-corrected dimensional splitting and permits stable semi-Lagrangian integrations using CFL numbers larger than one. Two methods for monotonicity preservation are used here: flux correction and modification of the underlying polynomial reconstruction.

Introduction

The accurate simulation of the effectively inviscid transport of scalar fields such as temperature or the concentrations of chemical species plays an important role in the modeling of many physical systems. The inviscid transport of a conservative scalar may be described by the conservation law

$$\frac{\mathrm{\partial }\rho \psi }{\mathrm{\partial }t}+\nabla ·(\rho \mathbf{u}\psi )=0\text{,}$$

where $\rho$, $\psi$ and $\mathbf{u}$ are the density, scalar concentration and vector velocity fields, respectively. For simplicity, we ignore any source and sink terms related to chemical reactions or changes in phase that might appear in more general applications and assume those processes are integrated in a separate fractional step. Note, however, that such processes may give rise to sharp gradients in $\psi$ that must be handled accurately in the approximation of (1).

It has long been recognized that numerical approximations to (1) often require a method that is monotonicity preserving in the vicinity of discontinuities and poorly resolved gradients, but switches to a higher-order, less diffusive formulation where the solution is smooth. One of the first techniques proposed for constructing such a scheme is flux-corrected transport (FCT) [1]. The general FCT strategy [2] is to compute approximations to the true fluxes $\rho \mathbf{u}\psi$ using both a low-order monotone method and a higher-order scheme. In regions where there is no possibility of generating new maxima and minima, the solution is updated by computing the divergence of the high-order fluxes. In other regions, the high-order fluxes are “corrected” to prevent the development of new maxima and minima. One systematic problem with flux correction algorithms is that they erroneously damp smooth extrema. As the phase of a wave shifts and its peak translates between a pair of grid points over an interval longer than a single time step, there should be time steps in which the maximum grid-point value of the function increases (for example, if the peak of a continuous function shifts from a position midway between two grid points to exactly coincide with one of the grid points). Flux correction algorithms do not distinguish between increases in the maximum of the grid-point values produced by the translation of a wave crest and spurious increases created by non-monotone schemes in the vicinity of poorly resolved gradients. Furthermore, in many applications, only a small percentage of the grid cells in the entire domain actually require flux correction or flux limiting, and throughout the remainder of the domain, significant computational effort is expended to apply a flux-limiting algorithm that is either unnecessary or detrimental.

Similar problems with the damping of smooth extrema are also encountered with a related class of flux-limiter and slope-limiter methods in which the magnitude of the high-order flux is “limited” to keep the solution total variation diminishing (TVD) [3], [4], [5]. Higher-order generalizations of the slope-limiter approach, in which piecewise parabolas are fitted to finite-volume averages of $\psi$ to estimate fluxes at the cell interfaces and these parabolas are subsequently modified to ensure monotonicity preservation (PPM [6], [7]), continue to suffer the same type of spurious damping.

The damping of smooth well-resolved extrema may be avoided using WENO methods [8], [9]. WENO methods estimate the fluxes in (1) using polynomial approximations to $\rho \psi$ of identical order spanning different sets of grid cells and compute the actual flux as a weighted average of the fluxes given by each of these polynomial approximations. The weights are determined by the local smoothness of the polynomial approximation over each stencil such that those regions which include sharp, poorly resolved gradients receive negligible weighting. In smooth regions, the weights are computed to yield the highest possible order of accuracy.

WENO methods have been used with considerable success in a wide variety of applications, including those considered here. Nevertheless in our tests, solutions to (1) obtained using a straight-forward implementation of the fifth-order WENO approximation [9] required much more computational time than an FCT approximation of roughly similar accuracy. WENO methods also suffer from a tendency for the weights of the various stencils to revert rather slowly to the underlying high-order method as the numerical resolution of a smooth function improves [10].

Here we propose a hybrid method in which monotonicity preservation is enforced only in those regions where simple WENO-motivated smoothness criteria indicate a poorly resolved steep gradient. As will be demonstrated by several test problems in Section 4, two different implementations of this approach using PPM for the underlying advection scheme preserve smooth extrema while minimizing spurious overshoots and undershoots better than the conventional fifth-order WENO method, while requiring far less computation time. These hybrid methods can also be used in semi-Lagrangian formulations that enlarge the numerical domain of dependence to permit stable integrations at CFL numbers greater than one.

A variety of additional strategies have been proposed for detecting “troubled cells” where limiters should be applied to prevent spurious oscillations near a discontinuity while attempting to preserve smooth extrema [11]. Here we compare our WENO-motivated criteria to the troubled-cell identification methodology proposed by Zerroukat et al. [12], which is a refinement of criteria proposed earlier by Sun et al. [13] and Nair et al. [14].

In the following, the underlying advection scheme and approaches for monotonicity preservation, which include flux correction and modification of the underlying polynomial reconstruction, are described in detail in Section 2. The WENO-motivated criteria for selective monotonicity preservation is presented in Section 3. One and two-dimensional test problems comparing the various approaches are analyzed in Section 4, and Section 5 contains the conclusions.