Using the theoretical framework of algebraic flux correction and invariant domain preserving schemes, we introduce a monolithic approach to convex limiting in continuous finite element schemes for ...linear advection equations, nonlinear scalar conservation laws, and hyperbolic systems. In contrast to flux-corrected transport (FCT) algorithms that apply limited antidiffusive corrections to bound-preserving low-order solutions, our new limiting strategy exploits the fact that these solutions can be expressed as convex combinations of bar states belonging to a convex invariant set of physically admissible solutions. Each antidiffusive flux is limited in a way which guarantees that the associated bar state remains in the invariant set and preserves appropriate local bounds. There is no free parameter and no need for limit fluxes associated with the consistent mass matrix of time derivative term separately. Moreover, the steady-state limit of the nonlinear discrete problem is well defined and independent of the pseudo-time step. In the case study for the Euler equations, the components of the bar states are constrained sequentially to satisfy local maximum principles for the density, velocity, and specific total energy in addition to positivity preservation for the density and pressure. The results of numerical experiments for standard test problems illustrate the ability of built-in convex limiters to resolve steep fronts in a sharp and nonoscillatory manner.
•A new approach to algebraic flux correction for hyperbolic problems.•Parameter-free limiting of fluxes that produce bound-violating states.•Continuous dependence on the data, well-defined steady-state limit.•Convexity-based proofs of invariant domain preservation properties.•A tailor-made positivity-preserving limiter for the Euler equations.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this work, we discuss and develop multidimensional limiting techniques for discontinuous Galerkin (DG) discretizations of scalar hyperbolic problems. To ensure that each cell average satisfies a ...local discrete maximum principle (DMP), we impose inequality constraints on the local Lax–Friedrichs fluxes of a piecewise-linear (P1) approximation. Since the piecewise-constant (P0) version corresponds to a property-preserving low-order finite volume method, the validity of DMP conditions can always be enforced using slope and/or flux limiters. We show that the (currently rather uncommon) use of direct flux limiting makes it possible to construct more accurate DMP-satisfying approximations in which a weak form of slope limiting is used to prevent unbounded growth of solution gradients. After presenting two flux limiters that ensure the validity of local DMPs for cell averages, we discuss the design of slope limiters based on different kinds of inequality constraints. In particular, we derive new limiting procedures based on flux constraints and constraints for directional derivatives. The latter approach makes it possible to preserve directional monotonicity in applications to problems that require different treatment of different space directions. At the flux limiting stage, the anisotropy of the problem at hand can be taken into account by using a customized definition of local bounds for the DMP constraints. At the slope limiting stage, we adjust the magnitude of individual directional derivatives using low-order reconstructions from cell averages to define the bounds. In this way, we avoid unnecessary limiting of well-resolved derivatives at smooth peaks and in internal/boundary layers. The properties of selected algorithms are explored in numerical studies for DG-P1 discretizations of two-dimensional test problems. In the context of hp-adaptive DG methods, the new limiting procedures can be used in P1 subcells of macroelements marked as ‘troubled’ by a smoothness indicator.
•Unified framework for flux and slope limiting in DG methods.•New bound-preserving limiters for linear and nonlinear problems.•Proofs of local discrete maximum principles for cell averages.•Anisotropic limiting for fluxes and directional derivatives.•Entropy stabilization via flux limiting and slope penalization.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
A new approach to slope limiting for discontinuous Galerkin methods on arbitrary meshes is introduced. A local Taylor basis is employed to express the approximate solution in terms of cell averages ...and derivatives at cell centroids. In contrast to traditional slope limiting techniques, the upper and lower bounds for admissible variations are defined using the maxima/minima of centroid values over the set of elements meeting at a vertex. The correction factors are determined by a vertex-based counterpart of the Barth–Jespersen limiter. The coefficients in the Taylor series expansion are limited in a hierarchical manner, starting with the highest-order derivatives. The loss of accuracy at smooth extrema is avoided by taking the maximum of correction factors for derivatives of order
p
≥
1
and higher. No free parameters, oscillation detectors, or troubled cell markers are involved. Numerical examples are presented for 2D transport problems discretized using a DG method.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this work, we present a Flux-Corrected Transport (FCT) algorithm for enforcing discrete maximum principles in Radial Basis Function (RBF) generalized Finite Difference (FD) methods for ...convection-dominated problems. The algorithm is constructed to guarantee mass conservation and to preserve positivity of the solution for irregular data nodes. The method can be applied both for problems defined in a domain or if equipped with level set techniques, on a stationary manifold. We demonstrate the numerical behavior of the method by performing numerical tests for the solid-body rotation benchmark in a unit square and for a transport problem along a curve implicitly prescribed by a level set function. Extension of the proposed method to higher dimensions is straightforward and easily realizable.
In this paper, we present a collection of algorithmic tools for constraining high-order discontinuous Galerkin (DG) approximations to hyperbolic conservation laws. We begin with a review of ...hierarchical slope limiting techniques for explicit DG methods. A new interpretation of these techniques leads to an unconditionally stable implicit algorithm for steady-state computations. The implicit global problem for the mean values (coarse scales) has the computational structure of a finite volume method. The constrained derivatives (fine scales) are obtained by solving small local problems. The interscale transfer operators provide a two-way iterative coupling between the solutions to the global and local problems. Another highlight of this paper is a new approach to compatible gradient limiting for the Euler equations of gas dynamics. After limiting the conserved quantities, the gradients of the velocity and energy density are constrained in a consistent manner. Numerical studies confirm the accuracy and robustness of the proposed algorithms.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
A new approach to the design of flux-corrected transport (FCT) algorithms for continuous (linear/multilinear) finite element approximations of convection-dominated transport problems is pursued. The ...algebraic flux correction paradigm is revisited, and a family of nonlinear high-resolution schemes based on Zalesak’s fully multidimensional flux limiter is considered. In order to reduce the cost of flux correction, the raw antidiffusive fluxes are linearized about an auxiliary solution computed by a high- or low-order scheme. By virtue of this linearization, the costly computation of solution-dependent correction factors is to be performed just once per time step, and there is no need for iterative defect correction if the governing equation is linear. A predictor–corrector algorithm is proposed as an alternative to the hybridization of high- and low-order fluxes. Three FEM-FCT schemes based on the Runge–Kutta, Crank–Nicolson, and backward Euler time-stepping are introduced. A detailed comparative study is performed for linear convection–diffusion equations.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
Abstract
The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy ...stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.
This work introduces a new type of constrained algebraic stabilization for continuous piecewise-linear finite element approximations to the equations of ideal magnetohydrodynamics (MHD). At the first ...step of the proposed flux-corrected transport (FCT) algorithm, the Galerkin element matrices are modified by adding graph viscosity proportional to the fastest characteristic wave speed. At the second step, limited antidiffusive corrections are applied and divergence cleaning is performed for the magnetic field. The limiting procedure developed for this stage is designed to enforce local maximum principles, as well as positivity preservation for the density and thermodynamic pressure. Additionally, it adjusts the magnetic field in a way which penalizes divergence errors without violating conservation laws or positivity constraints. Numerical studies for 2D test problems are performed to demonstrate the ability of the proposed algorithms to accomplish this task in applications to ideal MHD benchmarks.
•Localized element-based FCT algorithm for continuous finite element discretizations of the MHD system.•Synchronized and sequential a priori limiters based on local maximum principles for control variables.•Synchronized a posteriori limiting to enforce positivity preservation for the thermodynamic pressure.•Built-in penalization of divergence errors without violating maximum principles or conservation laws.
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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 the number of flux potentials is less than the number of fluxes in the multidimensional case, the potential-based version of optimal flux control involves fewer unknowns than direct calculation of optimal fluxes. We show that the feasible set of a potential-state potential-target (PP) optimization problem is nonempty and choose a primal–dual Newton method for calculating the optimal flux potentials. The results of numerical studies for linear and nonlinear conservation laws in 2D demonstrate the superiority of the new OB-PP algorithms to closed-form flux limiting under worst-case assumptions.
•Enforcement of maximum principles using flux-type optimal control.•Formulation of optimization problems in terms of flux potentials.•Applicability to spatial semi-discretizations and fully discrete schemes.•Possibility of enforcing constraints that are violated by exact solutions.•Existence proof based on bound-preserving residual distribution.•Solution of nonlinear optimization problems using a barrier method.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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