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Bound-preserving Flux Limiting for High-Order Explicit Runge–Kutta Time Discretizations of Hyperbolic Conservation Laws

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Abstract

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 recognize undershoots/overshoots, as well as 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. In this context, we enforce global bounds and prove preservation of accuracy for the linear advection equation.

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Data Availability

The code to reproduce the datasets (in all tables) is available at https://github.com/manuel-quezada/BP_Lim_for_RK_Methods.

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Funding

The work of Dmitri Kuzmin and Johanna Grüll was supported by the German Research Association (DFG) under Grant KU 1530/23-1. The work of Manuel Quezada de Luna and David I. Ketcheson was funded by King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia.

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Authors and Affiliations

  1. Institute of Applied Mathematics (LS III), TU Dortmund University, Vogelpothsweg 87, D-44227, Dortmund, Germany

    Dmitri Kuzmin & Johanna Grüll

  2. King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

    Manuel Quezada de Luna & David I. Ketcheson

Authors
  1. Dmitri Kuzmin
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  2. Manuel Quezada de Luna
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  3. David I. Ketcheson
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  4. Johanna Grüll
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Correspondence to Manuel Quezada de Luna.

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The code to reproduce the numerical experiments is available at https://github.com/manuel-quezada/BP_Lim_for_RK_Methods.

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Appendix A. High-Order Baseline RK Methods

Appendix A. High-Order Baseline RK Methods

In this appendix, we provide details of the three high-order explicit baseline RK methods for solving

$$\begin{aligned} \frac{\mathrm d u_i}{\mathrm d t} = F_i({\hat{u}}):=-\frac{1}{|K_i|}\sum _{j\in {\mathcal {N}}_i}|S_{ij}|H_{ij}({\hat{u}}). \end{aligned}$$
(A.1)

1.1 Appendix A.1. Fourth-Order Strong Stability Preserving (SSP54) RK Method

The SSP-RK time integrator that we consider in this work is the 4th-order method proposed in [47, 48]. The Butcher form of its intermediate stages is as follows:

$$\begin{aligned}&y^{(1)} = u^n + 0.391752226571890\varDelta t F(u^n), \\&\quad y^{(2)} = 0.444370493651235u^n + 0.555629506348765 y^{(1)}\\&\qquad \quad +0.368410593050371\varDelta t F(y^{(1)}), \\&\quad y^{(3)} = 0.620101851488403 u^n + 0.379898148511597 y^{(2)}\\&\qquad \quad + 0.251891774271694\varDelta tF(y^{(2)}), \\&\quad y^{(4)} = 0.178079954393132 u^n + 0.821920045606868 y^{(3)}\\&\qquad \quad + 0.544974750228521 \varDelta t F(y^{(3)}), \\&\quad y^{(5)} = 0.517231671970585 y^{(2)} + 0.096059710526147 y^{(3)} \\&\qquad \quad + 0.063692468666290 \varDelta t F(y^{(3)}) \\&\quad + 0.386708617503269 y^{(4)} + 0.226007483236906 \varDelta t F(y^{(4)}). \end{aligned}$$

Note that each stage is a convex combination of Euler steps. Therefore, if \(F(\cdot )\) is a BP spatial discretization and \(u^n\) is BP, then each stage is BP under appropriate time step restrictions. The RK update is given by \(u^{n+1} = y^{(5)}\). Hence, if the stages are BP, \(u^{n+1}\) is BP and no extra limiting is needed.

1.2 Appendix A.2. Fifth-Order Extrapolated Euler (ExE-RK5) RK Method

The Butcher tableau of the 5th-order extrapolated Euler RK method is given by

$$\begin{aligned} \begin{array}{c|ccccccccccc} 0 &{} 0 &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ 1/2 &{} 1/2 &{} 0 &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ 1/3 &{} 1/3 &{} 0 &{} 0 &{} &{} &{} &{} &{} &{} &{} &{} \\ 2/3 &{} 1/3 &{} 0 &{} 1/3 &{} 0 &{} &{} &{} &{} &{} &{} &{} \\ 1/4 &{} 1/4 &{} 0 &{} 0 &{} 0 &{} 0 &{} &{} &{} &{} &{} &{} \\ 1/2 &{} 1/4 &{} 0 &{} 0 &{} 0 &{} 1/4 &{} 0 &{} &{} &{} &{} &{} \\ 3/4 &{} 1/4 &{} 0 &{} 0 &{} 0 &{} 1/4 &{} 1/4 &{} 0 &{} &{} &{} &{} \\ 1/5 &{} 1/5 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} &{} &{} \\ 2/5 &{} 1/5 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/5 &{} 0 &{} &{} \\ 3/5 &{} 1/5 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/5 &{} 1/5 &{} 0 &{} \\ 4/5 &{} 1/5 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1/5 &{} 1/5 &{} 1/5 &{} 0 \\ \hline &{} 0 &{} -4/3 &{} 27/4 &{} 27/4 &{} -32/3 &{} -32/3 &{} -32/3 &{} 125/24 &{} 125/24 &{} 125/24 &{} 125/24. \\ \end{array} \end{aligned}$$
(A.2)

The intermediate stages (written in Shu-Osher and Butcher form) are as follows:

$$\begin{aligned} y^{(1)}&= u^n\approx & {} u(t^n), \\ y^{(2)}&= y^{(1)} + \frac{1}{2} \varDelta t F(y^{(1)})\approx & {} u(t^n+\varDelta t/2), \\ y^{(3)}&= y^{(1)} + \frac{1}{3} \varDelta t F(y^{(1)})\approx & {} u(t^n+\varDelta t/3), \\ y^{(4)}&= y^{(3)} + \frac{1}{3} \varDelta t F(y^{(3)}) = y^{(1)}+\frac{\varDelta t}{3} [F(y^{(1)}) + F(y^{(3)})]\approx & {} u(t^n+2\varDelta t/3), \\ y^{(5)}&= y^{(1)} + \frac{1}{4} \varDelta t F(y^{(1)})\approx & {} u(t^n+\varDelta t/4), \\ y^{(6)}&= y^{(5)} + \frac{1}{4} \varDelta t F(y^{(5)}) = y^{(1)}+\frac{\varDelta t}{4} [F(y^{(1)}) + F(y^{(5)})]\approx & {} u(t^n+\varDelta t/2), \\ y^{(7)}&= y^{(6)} + \frac{1}{4} \varDelta t F(y^{(6)}) = y^{(1)}+\frac{\varDelta t}{4} [F(y^{(1)}) + F(y^{(5)}) + F(y^{(6)})]\approx & {} u(t^n+3\varDelta t/4), \\ y^{(8)}&= y^{(1)} + \frac{1}{5} \varDelta t F(y^{(1)})\approx & {} u(t^n+\varDelta t/5), \\ y^{(9)}&= y^{(8)} + \frac{1}{5} \varDelta t F(y^{(8)}) = y^{(1)}+\frac{\varDelta t}{5} [F(y^{(1)}) + F(y^{(8)})]\approx & {} u(t^n+2\varDelta t/5), \\ y^{(10)}&= y^{(9)} + \frac{1}{5} \varDelta t F(y^{(9)}) = y^{(1)}+\frac{\varDelta t}{5} [F(y^{(1)}) + F(y^{(8)}) + F(y^{(9)})]\approx & {} u(t^n+3\varDelta t/5), \\ y^{(11)}&= y^{(10)} + \frac{1}{5} \varDelta t F(y^{(10)}) = y^{(1)}+\frac{\varDelta t}{5} [F(y^{(1)}) + F(y^{(8)}) + F(y^{(9)}) + F(y^{(10)})]\approx & {} u(t^n+4\varDelta t/5). \end{aligned}$$

Note that if \(F(\cdot )\) is a BP spatial discretization and \( u^n\) is BP, then each stage of this ExE-RK method is BP under appropriate time step restrictions. The approximations \(y^{(1)}\), \(y^{(2)}\), \(y^{(3)}\), \(y^{(5)}\), and \(y^{(8)}\) are BP because they correspond to forward Euler updates of \(u^n\). The remaining stages are BP since \(y^{(m)}\) is a forward Euler update of a BP approximation \(y^{(r)}\) for some \(r\in \{1,\ldots ,m-1\}\).

The Aitken–Neville interpolation yields the temporally 5th-order approximation

$$\begin{aligned} u^\mathrm{RK} =&\frac{1}{24}\left[ y^{(1)}+\varDelta t F(y^{(1)})\right] -\frac{8}{3}\left[ y^{(2)}+\frac{1}{2}\varDelta t F(y^{(2)})\right] +\frac{81}{4}\left[ y^{(4)}+\frac{1}{3}\varDelta t F(y^{(4)})\right] \\&-\frac{128}{3}\left[ y^{(7)}+\frac{1}{4}\varDelta t F(y^{(7)})\right] +\frac{625}{24}\left[ y^{(11)}+\frac{1}{5}\varDelta t F(y^{(11)})\right] . \end{aligned}$$

Note that this Euler extrapolation method combines \(S=5\) first-order approximations of \(u^{n+1}\). Since this combination is not convex, \(u^\mathrm{RK}\) is not necessarily BP even if \(y^{(1)},\ldots ,y^{(11)}\) are BP. To enforce the BP property of the final solution, we perform flux limiting using the Butcher form representation

$$\begin{aligned} u^\mathrm{RK}&= u^n + \varDelta t \Big [ -\frac{4}{3}F(y^{(2)}) + \frac{27}{4}F(y^{(3)}) + \frac{27}{4}F(y^{(4)}) - \frac{32}{3}F(y^{(5)}) - \frac{32}{3}F(y^{(6)}) \nonumber \\&\quad -\frac{32}{3}F(y^{(7)})+ \frac{125}{24}F(y^{(8)}) + \frac{125}{24}F(y^{(9)}) + \frac{125}{24}F(y^{(10)} + \frac{125}{24}F(y^{(11)}) \Big ]. \end{aligned}$$
(A.3)

1.3 Appendix A.3. Sixth-Order (RK76) RK Method

This 6th-order RK method, proposed in [49], consists of seven stages and has the Butcher tableau

$$\begin{aligned} \begin{array}{c|ccccccc} 0 &{} 0 &{} &{} &{} &{} &{} &{} \\ 1/3 &{} 1/3 &{} 0 &{} &{} &{} &{} &{} \\ 2/3 &{} 0 &{} 2/3 &{} 0 &{} &{} &{} &{} \\ 1/3 &{} 1/12 &{} 1/3 &{} -1/12 &{} 0 &{} &{} &{} \\ 1/2 &{} -1/16 &{} 9/8 &{} -3/16 &{} -3/8 &{} 0 &{} &{} \\ 1/2 &{} 0 &{} 9/8 &{} -3/8 &{} -3/4 &{} 1/2 &{} 0 &{} \\ 1 &{} 9/44 &{} -9/11 &{} 63/44 &{} 18/11 &{} 0 &{} -16/11 &{} 0\\ \hline &{} 11/120 &{} 0 &{} 27/40 &{} 27/40 &{} -4/15 &{} -4/15 &{} 11/120. \end{array} \end{aligned}$$
(A.4)

The intermediate stages of this method are not Euler steps. If we require them to be BP, the numerical fluxes \(H_{ij}\) should be constrained using the limiters from Sect. 3.3 in each stage. The BP property of the final solution \(u^\mathrm{RK}\) can be enforced similarly using the limiter from Sect. 3.2.

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Kuzmin, D., Quezada de Luna, M., Ketcheson, D.I. et al. Bound-preserving Flux Limiting for High-Order Explicit Runge–Kutta Time Discretizations of Hyperbolic Conservation Laws. J Sci Comput 91, 21 (2022). https://doi.org/10.1007/s10915-022-01784-0

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