Flux-corrected transport (FCT) is one of the flux limiter methods. Unlike the total variation diminishing methods, obtaining the known FCT formulas for computing flux limiters is not quite transparent, and their transformation is not obvious when the original differential operator changes. We propose a novel formal mathematical approach to design flux correction for weighted hybrid difference schemes by using linear programming. The hybrid scheme is a linear combination of a monotone scheme and a high order scheme. The determination of maximal antidiffusive fluxes is treated as an optimization problem with a linear objective function. To obtain constraints for the optimization problem, inequalities that are valid for the monotone difference scheme are applied to the hybrid difference scheme. The numerical solution of the nonlinear optimization problem is reduced to the iterative solution of linear programming problems. A nontrivial approximate solution of the corresponding linear programming problem can be treated as the required flux limiters. We present flux correction formulas for scalar hyperbolic conservation laws and convection-diffusion equations. The designed flux-corrected transport for scalar hyperbolic conservation laws yields entropy solutions. Numerical results are presented.
We report in this paper the analysis for the linear and nonlinear version of the flux corrected transport (FEM-FCT) scheme in combination with the backward Euler time-stepping scheme applied to time-dependent convection-diffusion-reaction problems. We present the stability and error estimates for the linear and nonlinear FEM-FCT scheme. Numerical results confirm the theoretical predictions.
This is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection-diffusion equations [Appl. Math. Lett. \textbf{131} (2022) 108048] which focuses on high-dimensional linear/nonlinear cases under Dirichlet or Neumann boundary conditions. Several new corrected difference schemes are proposed based on the explicit Euler discretization in temporal derivative and central difference discretization in spatial derivatives. The priori estimate of the corrected scheme with application to constant convection coefficients is provided at lengt…
We provide a careful Fourier analysis of the Guermond-Pasquetti mass lumping correction technique [Guermond J.-L., Pasquetti R. A correction technique for the dispersive effects of mass lumping for transport problems. - Computer Methods in Applied Mechanics and Engineering. - 2013. - Vol. 253. - P. 186-198] applied to pure transport and convection-diffusion problems. In particular, it is found that increasing the number of corrections reduces the accuracy for problems with diffusion; however all the corrected schemes are more accurate than the consistent Galerkin formulation in this case. For …
In this study we consider unconditionally non-oscillatory, high order implicit time marching based on time-limiters. The first aspect of our work is to propose the high resolution Limited-DIRK3 (L-DIRK3) scheme for conservation laws and convection-diffusion equations in the method-of-lines framework. The scheme can be used in conjunction with an arbitrary high order spatial discretization scheme such as 5th order WENO scheme. It can be shown that the strongly S-stable DIRK3 scheme is not SSP and may introduce strong oscillations under large time step. To overcome the oscillatory nature of DIRK…
A compact fourth-order doubly conservative active flux method with maximum-principle-preserving limiting for degenerate convection--diffusion equations
School of Science and Engineering, The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳）理工学院）
Convection schemes are a large source of error in global weather and climate models, and modern resolutions are often too fine to parameterise convection but are still too coarse to fully resolve it. Recently, numerical solutions of multi-fluid equations have been proposed for a more flexible and consistent treatment of sub-grid scale convection, including net mass transport by convection and non-equilibrium dynamics. The technique involves splitting the atmosphere into multiple fluids. For example, the atmosphere could be divided into buoyant updrafts and stable regions. The fluids interact t…
A lattice Boltzmann scheme that is close to pure upwind, low diffusive and entropy satisfying Engquist-Osher scheme has been formulated for hyperbolic scalar conservation laws. A model for source terms, with scalar conservation laws, is introduced to remove the numerical imbalance between convection and source terms.
We prove in this paper the weak consistency of a general finite volume convection operator acting on discrete functions which are possibly not piecewise-constant over the cells of the mesh and over the time steps. It yields an extension of the Lax-Wendroff if-theorem for general colocated or non-colocated schemes. This result is obtained for general polygonal or polyhedral meshes, under assumptions which, for usual practical cases, essentially boil down to a flux-consistency constraint; this latter is, up to our knowledge, novel and compares the discrete flux at a face to the mean value over t…
An posteriori error estimator for discontinuous Galerkin discretisations of convection-diffusion problems with application to Earth's mantle convection simulations
It is known that Flux Corrected Transport algorithms can produce entropy-violating solutions of hyperbolic conservation laws. Our purpose is to design flux correction with maximal antidiffusive fluxes to obtain entropy solutions of scalar hyperbolic conservation laws. To do this we consider a hybrid difference scheme that is a linear combination of a monotone scheme and a scheme of high-order accuracy. Flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the monotone sch…
To ensure preservation of local or global bounds for numerical solutions of conservation laws, we constrain a baseline finite element discretization using optimization-based (OB) flux correction. The main novelty of the proposed methodology lies in the use of flux potentials as control variables and targets of inequality-constrained optimization problems for numerical fluxes. In contrast to optimal control via general source terms, the discrete conservation property of flux-corrected finite element approximations is guaranteed without the need to impose additional equality constraints. Since t…
In this paper, we develop high-order nodal discontinuous Galerkin (DG) methods for hyperbolic conservation laws that satisfy invariant domain preserving properties using a subcell flux corrections and convex limiting. These methods are based on a subcell flux corrected transport (FCT) methodology that involves blending a high-order target scheme with a robust, low-order invariant domain preserving method that is obtained using a graph viscosity technique. The new low-order discretizations are based on sparse stencils which do not increase with the polynomial degree of the high-order DG method.…
We introduce a general framework for enforcing local or global maximum principles in high-order space-time discretizations of a scalar hyperbolic conservation law. We begin with sufficient conditions for a space discretization to be bound preserving (BP) and satisfy a semi-discrete maximum principle. Next, we propose a global monolithic convex (GMC) flux limiter which has the structure of a flux-corrected transport (FCT) algorithm but is applicable to spatial semi-discretizations and ensures the BP property of the fully discrete scheme for strong stability preserving (SSP) Runge-Kutta time dis…
Invariant domain preserving limiting of time explicit and time implicit discretizations for systems of conservation laws
ONERA, Institut Polytechnique de Paris（法国航空航天研究院，巴黎理工学院）；MONHADE, Equipe Inria-ONERA, DAAA, ONERA, Institut Polytechnique de Paris（MONHADE，Inria-ONERA团队，DAAA，法国航空航天研究院，巴黎理工学院）
This work extends the concepts of algebraic flux correction and convex limiting to continuous high-order Bernstein finite element discretizations of scalar hyperbolic problems. Using an array of adjustable diffusive fluxes, the standard Galerkin approximation is transformed into a nonlinear high-resolution scheme which has the compact sparsity pattern of the piecewise-linear or multilinear subcell discretization. The representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under CFL-like conditions…
We propose a new family of high order staggered semi-implicit discontinuous Galerkin (DG) methods for the simulation of natural convection problems. Assuming small temperature fluctuations, the Boussinesq approximation is valid and the flow can simply be modeled by the incompressible Navier-Stokes equations coupled with a transport equation for the temperature and a buoyancy source term in the momentum equation. Our numerical scheme is developed starting from the work presented in [TD14], in which the spatial domain is discretized using a face-based staggered unstructured mesh. For the computa…
This paper presents the first systematic study on the fundamental problem of seeking optimal cell average decomposition (OCAD), which arises from constructing efficient high-order bound-preserving (BP) numerical methods within Zhang--Shu framework. Since proposed in 2010, Zhang--Shu framework has attracted extensive attention and been applied to developing many high-order BP discontinuous Galerkin and finite volume schemes for various hyperbolic equations. An essential ingredient in the framework is the decomposition of the cell averages of the numerical solution into a convex combination of t…
This paper is a rebuttal to the claim found in the literature that the MUSCL scheme cannot be third-order accurate for nonlinear conservation laws. We provide a rigorous proof for third-order accuracy of the MUSCL scheme based on a careful and detailed truncation error analysis. Throughout the analysis, the distinction between the cell average and the point value will be strictly made for the numerical solution as well as for the target operator. It is shown that the average of the solutions reconstructed at a face by Van Leer's kappa-scheme recovers a cubic solution exactly with kappa = 1/3, …
