We introduce a general framework for enforcing local or global inequality constraints in high-order time-stepping methods for a scalar hyperbolic conservation law. The proposed methodology blends an arbitrary Runge-Kutta scheme and a bound-preserving (BP) first-order approximation using two kinds of limiting techniques. The first one is a predictor-corrector method that belongs to the family of flux-corrected transport (FCT) algorithms. The second approach constrains the antidiffusive part of a high-order target scheme using a new globalized monolithic convex (GMC) limiter. The flux-corrected approximations are BP under the time step restriction of the forward Euler method in the explicit case and without any time step restrictions in the implicit case. The FCT and GMC limiters can be applied to antidiffusive fluxes of intermediate RK stages and/or of the final solution update. Stagewise limiting ensures the BP property of intermediate cell averages. If the calculation of high-order fluxes involves polynomial reconstructions from BP data, these reconstructions can be constrained using a slope limiter to correct unacceptable input. The BP property of the final solution is guaranteed for all flux-corrected methods . Numerical studies are performed for one-dimensional test problems discretized in space using explicit weighted essentially nonoscillatory (WENO) finite volume schemes .
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 discretizations. To circumvent the order barrier for SSP time integrators, we constrain the intermediate stages and/or the final stage of a general high-order RK method using GMC-type limiters. In this work, our theoretical and numerical studies are restricted to explicit schemes which are provably BP for sufficiently small time steps. The new GMC limiting framework offers the possibility of relaxing the bounds of inequality constraints to achieve higher accuracy at the cost of more stringent time step restrictions. The ability of the presented limiters to preserve global bounds and recognize well-resolved smooth solutions is verified numerically for three representative RK methods combined with weighted essentially nonoscillatory (WENO) finite volume space discretizations of linear and nonlinear test problems in 1D .
R
inequality constraints
maximum principles
R
time-stepping methods for
space-time discretizations of
R
The proposed methodology blends an arbitrary Runge-Kutta scheme and a bound-preserving
We begin with sufficient conditions for a space discretization to be bound preserving
R
first-order approximation using two kinds of limiting techniques. The first one is a predictor-corrector method that belongs to the family of
and satisfy a semi-discrete maximum principle. Next, we propose a global monolithic convex (GMC) flux limiter which has the structure of a
R
algorithms. The second approach constrains the antidiffusive part of a high-order target scheme using a new globalized monolithic convex (GMC) limiter. The flux-corrected approximations are BP under the time step restriction of the forward Euler method in the explicit case and without any time step restrictions in the implicit case. The FCT and GMC limiters can be applied to antidiffusive fluxes of intermediate RK
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 discretizations. To circumvent the order barrier for SSP time integrators, we constrain the intermediate
R
of the final solution update. Stagewise limiting ensures the BP property of intermediate cell averages. If the calculation of
the final stage of a general
R
fluxes involves polynomial reconstructions from BP data, these reconstructions can be constrained using a slope limiter to correct unacceptable input. The BP property of the final solution is guaranteed for all flux-corrected methods . Numerical studies are performed for one-dimensional test problems discretized in space using explicit
RK method using GMC-type limiters. In this work, our theoretical and numerical studies are restricted to explicit schemes which are provably BP for sufficiently small time steps. The new GMC limiting framework offers the possibility of relaxing the bounds of inequality constraints to achieve higher accuracy at the cost of more stringent time step restrictions. The ability of the presented limiters to preserve global bounds and recognize well-resolved smooth solutions is verified numerically for three representative RK methods combined with
R
schemes
space discretizations of linear and nonlinear test problems in 1D