Multidimensional Flux-corrected Transport Research Articles

Bound-preserving flux limiting schemes for DG discretizations of conservation laws with applications to the Cahn–Hilliard equation

Many mathematical models of computational fluid dynamics involve transport of conserved quantities, which must lie in a certain range to be physically meaningful. The analytical or numerical solution u of a scalar conservation law is said to be bound-preserving if global bounds u∗ and u∗ exist such that u∗≤u≤u∗ holds in the domain of definition. These bounds must be known a priori. To enforce such inequality constraints at least for element averages in the context of discontinuous Galerkin (DG) methods, the numerical fluxes must be defined and constrained in an appropriate manner. In this work, we introduce a general framework for calculating fluxes that produce non-oscillatory DG approximations and preserve relevant global bounds for element averages even if the analytical solution of the PDE violates them due to modeling errors. The proposed methodology is based on a combination of flux and slope limiting. The (optional) slope limiter adjusts the gradients to impose local bounds on pointwise values of the high-order DG solution, which is used to calculate the fluxes. The flux limiter constrains changes of element averages so as to prevent violations of global bounds. Since manipulations of the target flux may introduce a consistency error, it is essential to guarantee that physically admissible fluxes remain unchanged. We propose two kinds of flux limiters, which meet this requirement. The first one is of monolithic type and its time-implicit version requires the iterative solution of a nonlinear problem. Only a fully converged solution is provably bound-preserving. The time-explicit version of this limiter is subject to a time step restriction, which we derive in this article. The second limiter is an iterative version of the multidimensional flux-corrected transport (FCT) algorithm and works as postprocessed correction scheme. This fractional step limiting approach guarantees that each iterate is bound-preserving but avoidable consistency errors may occur if the iterative process is terminated too early. Each iterate depends only on local information of the previous iterate. This concept of limiting the numerical fluxes is also applicable to finite volume methods. Practical applicability of the proposed flux limiters as well as the benefits of using an optional slope limiter are demonstrated by numerical studies for the advection equation (hyperbolic, linear) and the Cahn–Hilliard equation (parabolic, nonlinear) for first-order polynomials. While both flux limiters work for arbitrary order polynomials, we discuss the construction of bound-preserving slope limiters, and show numerical studies only for first-order polynomials.

Synchronized flux limiting for gas dynamics variables

This work addresses the design of failsafe flux limiters for systems of conserved quantities and derived variables in numerical schemes for the equations of gas dynamics. Building on Zalesak's multidimensional flux-corrected transport (FCT) algorithm, we construct a new positivity-preserving limiter for the density, total energy, and pressure. The bounds for the underlying inequality constraints are designed to enforce local maximum principles in regions of strong density variations and become less restrictive in smooth regions. The proposed approach leads to closed-form expressions for the synchronized correction factors without the need to solve inequality-constrained optimization problems. A numerical study is performed for the compressible Euler equations discretized using a finite element based FCT scheme.

Twodee: The Health and Safety Laboratory's shallow layer model for heavy gas dispersion Part 2: Outline and validation of the computational scheme

Part 1 of this three part paper described the mathematical and physical basis of twodee, the Health and Safety Laboratory's shallow layer model for heavy gas dispersion. In this part, the numerical solution method used to simulate the twodee mathematical model is developed. The boundary conditions for the leading edge, discussed in part 1, make demanding requirements on the computational scheme used. The flux correction scheme of Zalesak [S.T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, Journal of Computational Physics, 31 (1979) 335–362] is used in twodee as this has all the required properties. The twodee code is then tested against a number of theoretical and computational benchmark problems.

Twodee: the Health and Safety Laboratory's shallow layer model for heavy gas dispersion Part 3: Experimental validation (Thorney Island)

Part 1 of this three-part paper described the mathematical and physical basis of twodee, the Health and Safety Laboratory's shallow layer model for heavy gas dispersion. In part 2, the numerical solution method used to simulate the twodee mathematical model was developed; the flux correction scheme of Zalesak [S.T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, Journal of Computational Physics, 31 (1979) 335–362.] was used in twodee. This paper compares results of the twodee model to the experimental results taken at Thorney Island [J. McQuaid, B. Roebuck, The dispersion of heavier-than-air gas from a fenced enclosure. Final report to the U.S. Coast Guard on contract with the Health and Safety Executive. Technical Report RPG 1185, Safety Engineering Laboratory, Research and Laboratory Services Division, Broad Lane, Sheffield S3 7HQ, UK, 1985.]. There is no evidence to suggest that twodee predictions could be improved by changing any of the entrainment parameters from generally accepted values [R.K.S. Hankin, Heavy gas dispersion over complex terrain, PhD thesis, Cambridge University, 1997.]. The twodee model was broadly insensitive to the exact values of the entrainment parameters.

Performance of some high accurate semi Lagrangian numerical schemes for the scalar advection equation

Most of the numerical schemes commonly used for solving the scalar advection equation are not simultaneously: positive definite, high accurate, mass conserving and monotone. Semi Lagrangian numerical schemes derived by van Leer [1977a, b] and Colella and Woodward [1984], which were developed for compressible flow problems, seem to have the potential in fulfilling these conditions, all at the same time. In order to assess their qualities for the scalar advection problems, three of the most accurate schemes have been selected and subjected to some standard test cases. The test results are compared with the test results obtained for two well known numerical schemes: the Second Moment Method (SMM) (Egan and Mahoney [1972]) and the Multidimensional Flux Corrected Transport (MFCT) (Zalesak [1979], [1981]). Before discussing the test results, the schemes are derived following a generalized approach.

Fully multidimensional flux-corrected transport algorithms for fluids

The theory of flux-corrected transport (FCT) developed by Boris and Book [ J. Comput. Phys. 11 (1973) 38; 18 (1975) 248; 20 (1976) 397] is placed in a simple, generalized format, and a new algorithm for implementing the critical flux limiting stage' in multidimensions without resort to time splitting is presented. The new flux limiting algorithm allows the use of FCT techniques in multidimensional fluid problems for which time splitting would produce unacceptable numerical results, such as those involving incompressible or nearly incompressible flow fields. The “clipping” problem associated with the original one dimensional flux limiter is also eliminated or alleviated. Test results and applications to a two dimensional fluid plasma problem are presented.