Abstract
We study a temporal step size control of explicit RungeKutta (RK) methods for compressible computational fluid dynamics (CFD), including the NavierStokes equations and hyperbolic systems of conservation laws such as the Euler equations. We demonstrate that errorbased approaches are convenient in a wide range of applications and compare them to more classical step size control based on a CourantFriedrichsLewy (CFL) number. Our numerical examples show that the errorbased step size control is easy to use, robust, and efficient, e.g., for (initial) transient periods, complex geometries, nonlinear shock capturing approaches, and schemes that use nonlinear entropy projections. We demonstrate these properties for problems ranging from wellunderstood academic test cases to industrially relevant largescale computations with two disjoint code bases, the open source Julia packages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.
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1 Introduction
Hyperbolic conservation laws are at the heart of many physical models in science and engineering, e.g., fluid dynamics describing the airflow around an airplane, acoustics, and space plasma modeling. Usually, the systems need to be solved numerically. A common approach is to apply the method of lines, i.e., to apply a spatial semidiscretization first and solve the resulting ordinary differential equation (ODE) numerically. For this task, explicit RungeKutta (RK) methods are among the most commonly used schemes because of their efficiency and parallel scalability [48, 53, 55].
Since demanding simulations require a significant amount of compute resources, there is a strong interest in developing efficient numerical methods. Here, the efficiency can be measured by the required compute time, energy, and human time. The time and energytosolution depend on the algorithms and their implementation as well as the hardware [4, 41, 93] and are often a focus of high performance computing (HPC) studies. The required human time depends on the robustness of the overall numerical method and the sensitivity with respect to parameters that may need to be tuned to ensure the robustness and/or the efficiency. In this article, we study an errorbased step size control for compressible flow problems to demonstrate that it is efficient in all three aspects.
We use entropydissipative semidiscretizations to ensure the robustness [29, 31, 83, 86] but do not investigate specific implementation techniques discussed elsewhere [33, 61, 77]. Instead, we focus on the step size control of time integration methods applied to ODEs resulting from spatial semidiscretizations of compressible flow problems. The goal is to adapt the time step size \(\Delta t\) such that it is as big as possible while still satisfying the accuracy and stability requirements. For explicit RK methods applied to (highorder) semidiscretizations, the stability requirements are usually more restrictive. Thus, a commonly used approach is to adapt the time step size according to a CourantFriedrichsLewy (CFL) number chosen by the user. This involves estimating local wave speeds and mesh spacing, which can be demanding for complex systems and curved highorder meshes in multiple space dimensions required in practice. The optimal CFL number depends on the space and time discretizations and possibly on the specific problem [14, 58, 84]. An alternative, widely used in the context of generalpurpose ODE solvers, is an errorbased step size control. There, a local error estimate is obtained from an embedded method and fed into a controller adapting the time step size according to userspecified tolerances. This approach has also been used successfully for partial differential equations [6, 51, 74, 94] but appears to be lesswidely applied in the CFD community.
In this article, we systematically study an errorbased step size control for compressible CFD problems and compare it to CFLbased approaches. Specifically, we use the RK methods and controllers developed in [74] in two different software environments. Most of the examples use the open source packages OrdinaryDiffEq.jl [73] for ODE solvers and Trixi.jl [78, 85] for spatial semidiscretizations implemented in Julia [7]. Some largescale, industrially relevant CFD simulations are implemented in SSDC [67], which is built on top of the Portable and Extensible Toolkit for Scientific computing (PETSc) [3], its mesh topology abstraction (DMPLEX) [51], and its scalable ODE/DAE solver library [1]. All source code and instructions required to reproduce the numerical experiments using open source packages are available online in our reproducibility repository [79].
In the following, we briefly review RK methods and step size control techniques in Sect. 2. Afterwards, we study the robustness and the sensitivity to usersupplied parameters under a change of mesh structure of both step size control approaches in Sect. 3. Section 4 focuses on the effect of nonlinear shock capturing schemes. In Sect. 5, we discuss the impact of an initial transient period, e.g., in the cold start of a simulation initialized with free stream values. Thereafter, we study the step size control in the presence of a change of variables in Sect. 6. Next, we discuss the convenience of the errorbased step size control for exploratory research of new systems in Sect. 7. Finally, we summarize and discuss our findings in Sect. 8.
2 RungeKutta (RK) Methods and Step Size Control
We use the method of lines approach and first discretize a conservation law in space. In particular, we focus on spatial semidiscretizations using collocated discontinuous Galerkin (DG) methods (and related shock capturing schemes); see, e.g., [29, 31, 83, 86]. The resulting ODE
is then solved using numerical time integration methods. Here, we use explicit RK methods with an embedded error estimator of the form [12, 38]
Here, \(u^{n+1}\) is the numerical solution of the main method, \(\widehat{u}^{n+1}\) is the embedded methods solution used to obtain the local error estimate \(u^{n+1}  \widehat{u}^{n+1}\), and \(y_i\) are the stage values. RK methods are parameterized by their Butcher tableau
where \(A = (a_{ij})_{i,j=1}^s \in \mathbb {R}^{s \times s}\) is strictly lower triangular for explicit methods, \(b, c \in \mathbb {R}^s\), and \(\widehat{b}\in \mathbb {R}^{s+1}\). If \(\widehat{b}_{s+1} \ne 0\), the RK pair (2) uses the socalled firstsameaslast (FSAL) idea to include the predicted value at the new time in the embedded error estimator [27]. This can increase the performance of the embedded method and comes at no additional cost if the step is accepted, since \(f(t_{n+1}, u^{n+1})\) must be computed as the first stage of the next step anyway.
All methods considered in this article are applied in the local extrapolation mode, i.e., a main method of order q is coupled with an embedded error estimator of order \(\widehat{q}= q  1\). Whenever possible, we make use of lowstorage formats such as 3S*_{+} [50, 74]. We use the same naming convention of RK methods as in [74], i.e., NAMEq(\(\widehat{q}\))s indicates an sstage method of order q with the embedded error estimator of order \(\widehat{q}\). This base name is followed by additional information on lowstorage properties and a subscript “F” for FSAL methods.
2.1 Time Integration Loop and Callbacks
We use implementations of the time integration methods available in OrdinaryDiffEq.jl [73] in Julia or PETSc/TS [1]. The basic steps for the time integration loop are given in Algorithm 1. At first, a preliminary time step is performed with the given RK method. If the errorbased step size control is activated, the embedded error estimator is also computed; it is used to update the time step size and determine whether the time step is rejected. After accepting a time step, callbacks are activated. We use these callback mechanisms for adaptive mesh refinement (AMR) and, if activated, CFLbased step size control.
A more detailed overview of the time integration loop is given below in Algorithm 2, discussed after introducing the step size control mechanisms.
2.2 CFLBased Step Size Control
Explicit time integration methods for firstorder hyperbolic conservation laws are subject to a CFL time step restriction of the form \(\Delta t\leqslant C \Delta x/ \lambda _\textrm{max}\) [22]. However, it is nontrivial to provide sharp estimates of the terms in this restriction, e.g., an appropriate value for \(\Delta x\) on curved meshes for highorder DG methods in multiple space dimensions. In general, the CFLbased step size control requires user estimates of the maximal local wave speeds \(\lambda _\textrm{max}(u^n_i)\) at the degree of freedom \(u^n_i\) and the corresponding local mesh spacing \(\Delta x_i\). Then, the time step is determined as
where \(\nu\) is the desired CFL number. For DG methods using polynomials of degree \(p\) on (possibly) curved grids for the linear advection equation \(\partial _t u + (a \cdot \partial _x) u = 0\), we follow [2, 74] and use the estimate
where \(d\) is the number of spatial dimensions, \(J_i\) the determinant of the grid Jacobian \(\partial _{x} \xi\) at node i, \((J \partial _{x} \xi ^j)_i\) the contravariant basis vector in direction j at node i [54, Chapter 6], and 2 is the size of the reference element. For nonlinear conservation laws, a is replaced by appropriate estimates of the local wave speeds.
2.3 ErrorBased Step Size Control
We perform the errorbased step size control based on controller ideas from digital signal processing [36, 37, 88–90]. In particular, we use PID controllers that select a new time step based on
where \(k = \min (q, \widehat{q}) + 1\), q is the order of the main method, and \(\widehat{q}\) is the order of the embedded method. Because \(\widehat{q}= q  1\) in our case, we have that \(k = \widehat{q}+ 1 = q\). Moreover, \(\beta _i\) are the controller parameters, \(\kappa\) is a step size limiter, and
where m is the total number of degrees of freedom in u, and \(\tau _a, \tau _r\) are the absolute and relative error tolerances. In OrdinaryDiffEq.jl, \(\widetilde{u} = u^{n}\), while PETSc uses \(\widetilde{u} = \widehat{u}^{n+1}\). Unless stated otherwise, we use equal tolerances \(\tau _a= \tau _r= \texttt {tol}\) and the default step size limiter \(\kappa (a) = 1 + \arctan (a  1)\) [91].
The decision whether the step shall be accepted or rejected is determined by the size of the factor multiplying the time step size \(\Delta t_{n}\) in (6). The default option used for all numerical experiments is to accept a step whenever the limited factor is at least \(0.9^2\). Otherwise, the step is rejected and the time step size \(\Delta t_{n}\) is set as the new predicted step size (6). Another common strategy is to decide whether a step should be rejected based only on the current local error estimate. Söderlind and Wang [89, 91] argued why the approach we described here can be beneficial to reduce the amount of (unnecessary but expensive) step rejections.
To make the step size control fully automatic, we use the estimate of the initial step size implemented in OrdinaryDiffEq.jl from the algorithm described by Hairer et al. [38, p. 169]. \(\epsilon _{n}\) and \(\epsilon _{n1}\) are initially set to one (for \(n = 0\)).
A more detailed overview of the time integration loop including additional aspects about the step size control and step rejections is shown in Algorithm 2. Parameters such as the threshold \(0.9^2\) used to determine whether a step should be accepted or the limiting threshold \(w_\textrm{min}\) used in Algorithm 2 are based on recommendations in reference works such as [38, Sect. IV.2] and [12, Sect. 371]; we always use their default values and do not consider them as userfacing parameters that need to be chosen manually.
2.4 Importance of Controller Parameters
The RK pair (2) and the PID controller (6) must be developed together to obtain good results. In particular, semidiscretizations of conservation laws limited by stability should be integrated using a combination of RK pair and controller leading to a stable step size control, based on the test equation [40]; see also [39, Sect. IV.2] and [49]. This has been demonstrated for spectral element discretizations for compressible flows in [74]. In this article, we use the optimized controllers of [74].
2.5 Representative RungeKutta (RK) Methods Used in This Article
There is a vast amount of literature on RK methods. Many schemes have been designed as generalpurpose methods for low or hightolerance applications or even specifically for hyperbolic conservation laws. While we have performed numerical experiments with various schemes, we restrict this article to the following representative set of methods. First, we consider the generalpurpose method:

\({\textrm{BS}}{3}{(2)}_{\textrm{F}}\), the thirdorder, threestage method of [8] (BS3() in OrdinaryDiffEq.jl, 3bs in PETSc).
This method has been shown to be among the most efficient—if not the most efficient—generalpurpose method for the CFD problems in which we are interested [74]. Next, we consider methods optimized for spectral element semidiscretizations of compressible CFD, namely,

\({\textrm{RDPK}}{3}(2){5}_\textrm{F}[3{\textrm{S}^*}_{+}]\), the thirdorder, fivestage method of [74] (RDPK3SpFSAL35() in OrdinaryDiffEq.jl);

\({\textrm{RDPK}}{4}(3){9}_\textrm{F}[3{\textrm{S}^*}_{+}]\), the fourthorder, ninestage method of [74] (RDPK3SpFSAL49() in OrdinaryDiffEq.jl).
Finally, we consider the strong stability preserving (SSP) method:
3 Robustness Under Change of Mesh Structure
Both CFL and errorbased step size controls come with parameters that must be chosen by the user, either a CFL factor \(\nu\) or a tolerance \(\texttt {tol}\). However, the sensitivity with respect to these parameters differs significantly. The CFL factor \(\nu\) influences the time step sizes—and thus the efficiency—linearly, while the tolerance \(\texttt {tol}\) has roughly a logarithmic influence. Thus, it is arguably easier to choose an appropriate tolerance than an optimal CFL factor.
Moreover, in the stability limited regime, it is often convenient to use the errorbased step size control, since there is usually a range of tolerances resulting in the same number of righthand side (RHS) evaluations a manually optimized CFLcontrol could achieve at best. Furthermore, an optimal CFL factor \(\nu\) depends on the mesh. Thus, it can vary when introducing curved coordinates compared to a uniform grid. This was already demonstrated for a 2D linear advection equation in [74, Sect. 3]. Here, we extend this demonstration to the nonlinear compressible Euler equations of an ideal gas in \(d= 3\) space dimensions given by
where \(\rho\) is the density, v the velocity, \(\rho e\) the total energy, and p the pressure given by
with the ratio of specific heats \(\gamma = 1.4\). Specifically, we consider the classical inviscid TaylorGreen vortex with the initial data^{Footnote 1}
where \({Ma} = 0.1\) is the Mach number. We consider the domain \([\uppi , \uppi ]^3\) with periodic boundary conditions and meshes with 8 elements per coordinate direction. We apply entropydissipative DG methods with polynomials of degree \(p= 3\) using a local LaxFriedrichs flux at interfaces and the flux of [75, 80, 81] in the volume terms. We integrate the semidiscretizations in the time interval [0, 10].
We consider two meshes, a uniform Cartesian mesh and a curved mesh that heavily warps the Cartesian reference coordinates \((\xi , \eta , \zeta ) \in [1, 1]^3\) to the desired mesh in physical coordinates (x, y, z) with the mapping
which has been adapted from [17]. The curved mesh and the initial pressure are visualized in Fig. 1.
The required number of RHS evaluations for the selected, representative RK methods is shown in Fig. 2, for CFLbased and errorbased step size controls. All RK methods show the same qualitative behavior on both grids. The step size for this problem is largely limited by the stability, resulting in a range of tolerances yielding the optimal number of RHS evaluations one can also achieve by tuning the CFL factor manually (maximizing \(\nu\) to three significant digits so that the simulations do not crash). This range of tolerances spans up to five orders of magnitude. For looser tolerances, the simulations may crash due to positivity issues, typically in the first few time step. Stricter tolerances let the controller detect an accuracylimited regime and increase the number of RHS evaluations accordingly. This happens at looser tolerances for \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\) than for the other methods, since \({\textrm{SSP}}3(2)4[3{{\textrm{S}^{*}}}_{+}]\) is only optimized for the SSP coefficient, while the other methods are also constructed to minimize the principal truncation error. Since all methods are of the same order of accuracy, the slope of the increasing number of RHS evaluations is the same.
However, there is a significant difference between CFLbased and errorbased step size controls when switching the mesh types. Indeed, the CFL factor can be increased between 55% (for \({\textrm{BS}}{3(2)}{3}_{\textrm{F}}\)) and 74% (for \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\)) on the curved mesh compared to the equivalent Cartesian mesh. In particular, choosing an optimized CFL factor from the uniform mesh results in the same amount of efficiency loss when the CFL factor is not tuned again manually from scratch. In contrast, basically, the same range of tolerances can be used to get the optimal number of RHS evaluations on both grids for the errorbased step size control.
4 Change of Linear Stability Restrictions in Nonlinear Schemes
There are many shock capturing approaches for DG methods, e.g., artificial viscosity [35, 72], replacing DG elements by finite volume subcells [28, 87], modal filtering [63, 76], or specially constructed invariant domain preserving methods [34, 59, 71]. Here, we use the a priori convex blending of highorder DG elements with finite volume subcells described in [42].
4.1 Linear CFL Restrictions
Shock capturing approaches in nonlinear schemes will typically change the linear CFL restriction on the time step for explicit time integration schemes. Here, we demonstrate this for some thirdorder schemes of the representative RK methods introduced in Sect. 2.5.
We consider the linear advection equation
in the domain \([1, 1]^2\) with periodic boundary conditions and velocity \(a = (1, 1)^\text{T} / \sqrt{2}\). We discretize the domain using \(8^2\) uniform elements with polynomials of degree \(p= 3\) on LegendreGaussLobatto nodes. We choose a fixed blending parameter \(\alpha = 0.5\) in the shock capturing scheme of [42] with the standard DG collocation spectral element method (DGSEM), e.g., [54], and local LaxFriedrichs flux for the finite volume subcells. We compute the spectra of the standard DGSEM scheme and the shock capturing scheme. Due to the fixed choice of the blending parameter, both semidiscretizations are linear and we compute their spectra numerically.
The spectra embedded into the stability regions of the representative RK methods are visualized in Fig. 3. To make it easier to compare the RK methods, the stability regions are scaled by the effective number of RK stages, taking the FSAL property into account. The spectra of the semidiscretizations are scaled by the maximal factor, so that the standard DGSEM spectrum is in the region of absolute stability of the RK method. Clearly, the (scaled) spectra of the shock capturing semidiscretizations are partially outside of the stability regions. Thus, one can expect a more restrictive CFL condition than for the standard DGSEM schemes.
These linear CFL restrictions predict an approximately onequarter bigger CFL number for standard DGSEM discretizations compared to the shock capturing variants for \(\textrm{BS}{3}(2){3}_{\textrm{F}}\) and \(\textrm{RDPK}{3}(2){5}_{\textrm{F}}[{\mathrm{3\,S}^*}_{+}]\). For these time integration methods, the SC spectra extend significantly to the lefthalf of the complex plane outside of the stability regions, in particular near the negative real axis. The effect is qualitatively similar but quantitatively less pronounced for \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\) (ca. 5% instead of more than 20%).
4.2 Application to Nonlinear Magnetohydrodynamics: OrszagTang Vortex
Next, we apply the representative time integration methods to entropydissipative semidiscretizations of the ideal generalized Lagrange multiplier (GLM) magnetohydrodynamics (MHD) equations [9, 25] for the OrszagTang vortex setup [66]. Specifically, we use the initial condition
in the domain \([0, 1]^2\) with periodic boundary conditions, where \(\rho\) is the density, v the velocity, p the pressure, B the magnetic field, and \(\psi\) the Lagrange multiplier to control divergence errors. The ideal GLMMHD equations with ratio of specific heats \(\gamma = 5 / 3\) and divergence cleaning speed \(c_h = 1\) are discretized on a uniform mesh with \(2^6 = 64\) elements per coordinate direction and polynomials of degree \(p= 3\). We use the shock capturing method of [42] with the entropyconservative numerical flux of [44] and a local LaxFriedrichs flux at interfaces and for finite volume subcells, both with the nonconservative Powell source term. We use the product of density and pressure as the shock indicator variable with blending parameters \(\alpha _{\textrm{max}} = 0.5\) and \(\alpha _{\textrm{min}} = 0.001\).
As shown in Fig. 4, the flow and its numerical approximation remain smooth in its initial phase. Between \(t = 0.1\) and \(t = 0.2\), the shock capturing indicator detects troubled cells and activates the finite volume shock capturing mechanism. At the final time, several shocklets are visible in the approximation.
We recorded the time step sizes and effective CFL numbers, i.e., the CFL factor computed a posteriori based on the time step size chosen by the errorbased step size controller, after every accepted time step for some representative RK methods. The tolerance is set to \(\texttt {tol}= 10^{4}\) in all cases with the errorbased step size control. For the CFLbased step size control, we bisected the maximal CFL number without blowup to three significant digits. The results are shown in Fig. 5. Clearly, the time step size of the CFLbased approaches does not vary significantly in time. Thus, the estimated CFL restriction is nearly constant. However, the nonlinear schemes activate shock capturing mechanisms between \(t = 0.1\) and \(t = 0.2\). Thus, the estimated linear CFL restriction should become more severe as discussed in Sect. 4.1. This can be observed also for the errorbased step size control; all RK methods use larger time steps initially until the shock capturing part is activated. Then, the errorbased step size control yields approximately the maximal step size also used by the CFLbased step size control with manually tuned CFL number. In particular, the initial time step size is approximately onequarter bigger with the errorbased step size control (after the controller has adapted the time step size from the automated initial guess). During the full simulation, the errorbased step size control results in the following performance improvements based on the number of function evaluations (see also Table 1): 13% for \(\textrm{BS}{3}(2){3}_{\textrm{F}}\), 11% for \({\textrm{RDPK}}{3}(2){5}_{\textrm{F}}[3{\textrm{S}^{*}}_{+}]\), and 6% for \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\).
While the quantitative numbers for \({\textrm{BS}}{3}(2){3}_{\textrm{F}}\) and \({\textrm{RDPK}}{3}(2){5}_{\textrm{F}}[3{{\textrm{S}}^*}_{+}]\) fit very well to the linear analysis, one could expect that the initial advantage of \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\) with the errorbased step size control should be smaller, since the stability region is less optimal for highorder DGSEM and relatively better suited for loworder shock capturing methods. First, we do not necessarily expect that such a linear analysis is quantitatively correct for general nonlinear problems, although it was shown to behave very well in applications [70]. Second, the shock capturing mechanism also changes the behavior of the ODE RHS, usually reducing the smoothness of the spatially discrete system, which also affects step size control.
As expected, the method \(\textrm{RDPK}{3}(2){5}_\textrm{F}[3{\textrm{S}^{*}}_+]\) optimized for spectral element discretizations performs better than the generalpurpose method \(\textrm{BS}{3}(2){3}_\textrm{F}\) and uses 15% less function evaluations (with the errorbased step size control). For this problem with a significant amount of shock capturing, \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\) with errorbased step size performs even better and uses additionally 16% less function evaluations than \(\textrm{RDPK}{3}(2){5}_\textrm{F}[3{\textrm{S}^{*}}_+]\).
In summary, the results of this section demonstrate that the errorbased step size control can react robustly and efficiently to varying linear CFL restrictions in nonlinear schemes. In particular, they usually require less parameter tuning while still achieving optimal performance. For this problem with a significant amount of shock capturing mechanisms, the strong stability preserving method \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\) performs well.
5 Initial Transient Period: Cold Start of a Simulation
When performing a cold start of a demanding CFD simulation, e.g., by initializing the flow around objects with free stream values, the initial transient period often requires smaller time steps than the fully developed simulation. For demanding simulations with positivity issues for highorder spatial schemes used in this section, we apply SSP time integration methods.
5.1 Double Mach Reflection of a Strong Shock
First, we consider the double Mach reflection of Woodward and Colella [97]. We set up an initial grid with 80 uniform elements that is adaptively refined during the simulation using p4est [11]. We use the shock capturing approach of [42] with DG elements using polynomials of degree \(p= 4\) and the entropyconservative numerical flux of [75, 80, 81] in the volume terms and a local LaxFriedrichs flux at interfaces and finite volume subcells. AMR is triggered every two time steps and the positivitypreserving limiter of [99] for density and pressure is applied after every RK stage. We apply \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\) to integrate the system in time \(t \in [0, 0.2]\). The complete setup can be found in the reproducibility repository [79]. Figure 6 shows the density and the mesh at the final time.
Figure 7 shows the evolution of the time step sizes and the effective CFL number. There is an initial transient period of ca. 100 time steps where both the time step size and the effective CFL number are much smaller than in the remaining simulation. This small initial step size is selected by the automatic detection algorithm and the errorbased step size control, and no manual intervention is necessary. The same simulation setup run with the CFLbased step size control crashes in the first few time steps, even with a small CFL number of \(\nu = 0.01\). The initial transient period can be captured with even smaller CFL numbers, but running the simulation afterwards in a reasonable amount of time required adapting the CFL factor. In contrast, the errorbased step size control can be used with the same parameters throughout the simulation.
5.2 Astrophysical Mach 2 000 Jet
Next, we consider a simulation of an astrophysical Mach 2 000 jet using the ideal compressible Euler equations with ratio of specific heat \(\gamma = 5 / 3\) based on the setup of [60]. Specifically, we use the initial condition
in the domain \([0.5, 0.5]\) equipped with periodic boundary conditions in the ydirection and Dirichlet boundary conditions using the initial data in the xdirection except for \(t > 0\), \(x =\)− 0.5, and \(y \in [\)− 0.05, 0.05], where we use boundary data (denoted by a superscript b)
We use an initially uniform mesh with 64 elements per coordinate direction and polynomials of degree \(p= 3\). We use the shock capturing approach of [42] with the entropyconservative numerical flux of [75, 80, 81] in the volume terms and a local LaxFriedrichs flux at interfaces and finite volume subcells. After every time step, we apply adaptive mesh refinement. Moreover, we apply the positivitypreserving limiter of [99] for density and pressure after every RK stage. We integrate the semidiscrete system in time using \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\) for \(t \in [0, 10^{3}]\). The complete setup can be found in the reproducibility repository [79].
Figure 8 shows a snapshot of the numerical solution and the adapted grid at the final time \(t = 10^{3}\). While the initial condition is given by uniform free stream values, the boundary condition (15) changes the numerical approximation rapidly in time. Consequently, the mesh is refined over time and adapts to the solution structures.
Figure 9 shows the evolution of the time step for this demanding simulation. The initial time step must be very small—of the order \(\Delta t\approx 10^{12}\)—to allow the simulation to run. For this setup, we set \(\Delta t= 10^{12}\) initially; other choices are also possible and are adapted to similar values automatically by the controller, but the fully automatic initial guess is not sufficient in this case.
The initial transient period lasts for ca. 100 time steps where the errorbased step size controller increases the time step size gradually. Afterwards, the time step size plateaus. Handling such an initial transient period is difficult for standard CFLbased step size controllers. Initially, a CFL factor of \(\nu \approx 10^{9}\) is required; \(\nu = 10^{8}\) results in a blowup of the simulation. One could implement a more complicated CFLbased controller adapting the CFL factor in the initial transient period or restart the simulation after ca. 100 time steps with a larger CFL factor. However, no such additional techniques are required for the errorbased step size control, making it easier to use in this case.
5.3 Delta Wing
The next test case setup approximates the solution of the compressible NavierStokes equations. We consider an industrially relevant simulation of a 65\(^\circ\) swept delta wing at a Mach number \({Ma} = 0.9\), a Reynolds number \({Re}=10^{6}\) (based on the mean aerodynamic chord), and setting the angle of attack to \(\textrm{AoA}= {13}^\circ\). The geometry was proposed by Hummel and Redeker [47] for the Second International Vortex Flow Experiment (VFE2) and manufactured based on an NASA wing geometry that served as reference configuration [20]. We consider the flow past the leadingedge configuration with a medium radius leading edge \(r_{\text{LE}}/\bar{c}=0.001\,5\), where \(\bar{c}={0.653}\,{\textrm{m}}\). The delta wing has a mean aerodynamic chord of \(\ell ={0.667}\,{\textrm{m}}\), a root chord length of \(c_{\text{r}} = 1.47 \ell\), and a wing span of \(b=1.37 \ell\). The sting present in the wind tunnel testing is kept downstream as part of the geometry up to the position \(x_1/c_{\text{r}}=1.758\), where the \(x_1\) Cartesian coordinate points in the streamwise direction, as shown in the left part of Fig. 10.
For this test, we use the hpadaptive entropystable solver SSDC [67], which is able to deliver numerical results in good agreement with experimental data. Entropystable adiabatic noslip wall boundary conditions [23] are applied to the wing and sting surfaces, whereas freestream farfield boundary conditions are applied at the inlet and outlet planes. Due to the symmetry of the problem in the spanwise direction, half span of the flow is modeled through a symmetry boundary condition. On the rest of the boundary planes, entropystable inviscid wall boundary conditions [69] are prescribed. The grid consists of \(\approx 9.2\times 10^4\) hexahedral elements. It is subdivided into three blocks, as shown in the right part of Fig. 10, with each block corresponding to a different solution polynomial degree p. In particular, we use \(p=2\) in the farfield region, \(p=5\) in the region surrounding the delta wing and its support, and \(p=3\) elsewhere. Given the degree of the solution and the number of elements in each block, the total number of degrees of freedom (DOFs) is \(\approx 1.435 \times 10^{7}\).
The flow around a delta wing is peculiar. When the angle of attack exceeds 7\(^\circ\), typically, flow separation occurs at the leading edge. The rollup of the leadingedge vortices induces low pressure on the upper surface of the wing and enhances the lift. Figure 11 shows the instantaneous pressure coefficient distribution and streamlines colored by the velocity magnitude at \(t = 0.05\).
Results of simulations using some representative RK methods are shown in Fig. 12. For this cold start of the simulation with free stream values, all methods begin with a conservative estimate of the time step size. The errorbased step size controllers increase the time step size in the first few time steps and reach an asymptotically constant step size after a few hundred time steps.
The effective CFL number shown in Fig. 12 is basically proportional to the time step size itself. Running SSDC with the CFLbased step size control requires a CFL factor that is (at least initially)

13% smaller for \(\textrm{BS}{3}(2){3}_\textrm{F}\),

21% smaller for \(\textrm{RDPK}{3}(2){5}_\textrm{F}[3{\textrm{S}^{*}}_+]\),

32% smaller for \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\),
than the asymptotic CFL factor to avoid a blowup of the simulation. The required number of RHS evaluations and rejected steps are listed in Table 2.
5.4 NASA Common Research Model
The NASA common research model (CRM) was conceived in 2007. Its aerodynamic design was completed in 2008, responding to needs broadly expressed within the US and international aeronautics communities for modern/industryrelevant geometries coupled with advanced experimental data for applied computational fluid dynamic validation studies [82]. Here, we consider the compressible NavierStokes equations for a flow over the NASA CRM at an angle of attack of 10\(^\circ\), a Mach number \({Ma} = 0.85\), and a Reynolds number \({Re} = 5 \times 10^{6}\).
The computational domain contains \(12.8 \times 10^{6}\) hexahedral elements with a maximum aspect ratio of approximately 105. Entropystable adiabatic noslip wall boundary conditions [23] are applied to the aircraft, whereas freestream farfield boundary conditions are applied at the inlet and outlet planes. We set a solution polynomial degree \(p = 3\) in the whole domain. Given the degree of the solution and the number of cells, the number of DOFs is \(\approx 7.68 \times 10^{8}\). A zoom of the mesh near the surface of the right wing and the nacelle is shown in Fig. 13.
Results of simulations using some representative RK methods are shown in Fig. 14. Again, all methods use a conservative estimate of the initial time step size \(\Delta t\), which then increases \(\Delta t\) monotonically in the first few hundred time steps and reaches an asymptotically approximately constant step size afterwards. The effective CFL number shown in Fig. 14 is, again, basically proportional to the time step size itself. Running SSDC with the CFLbased step size control requires a CFL factor that is (at least initially)

16% smaller for \(\textrm{RDPK}{3}(2){5}_\textrm{F}[3{\textrm{S}^{*}}_+]\),

9% smaller for \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\),
than the asymptotic CFL factor to avoid a blowup of the simulation. For \(\textrm{BS}{3}(2){3}_\textrm{F}\), the asymptotic CFL factor from the errorbased step size control can be chosen for the complete simulation. The required number of RHS evaluations and rejected steps are listed in Table 3.
To conclude this test case, we highlight that the errorbased controller yields a substantially higher level of robustness for this industryrelevant test case under a change of mesh structure. To verify that, we simulate the same flow problem using the mesh illustrated in Fig. 15. This new mesh is one of the grids provided for the fifth highorder workshop held at Ceneaero, Belgium, and it corresponds to the mesh tagged as “CRMWBa2.75CoarseP2A”. When the CFLbased controller is used, the target CFL number is set to the effective asymptotic value obtained from the previous simulations with the mesh shown in Fig. 13, i.e., approximately 3.6 for all three RK methods (see Fig. 14). The simulations run successfully to the final time for all the three time integration methods tested, i.e., \(\textrm{BS}{3}(2){3}_\textrm{F}\), \(\textrm{RDPK}{3}(2){5}_\textrm{F}[3{\textrm{S}^{*}}_+]\), and \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\), when the errorbased time step controller is used. On the contrary, all simulations fail when we choose the CFLbased time step controller. The reason is simple, the effective asymptotic value of the CFL number for this new mesh is approximately a factor of three smaller than the value set, i.e., \(\approx 1.2\). This indicates that the CFLbased controller is not sufficiently robust under a change of mesh. In many cases, we could mitigate the instability by choosing a “very small value” for the target CFL number, but this would yield a very inefficient and computationally expensive time advancement of the simulation.
6 Change of Variables: Dissipative Methods Using an Entropy Projection
Whenever a semidiscretization is not strictly a collocation method, there are some transformations between the basic solution variables evolved in time and nodal values required to compute fluxes, e.g., methods using modal coefficients [16, 62], entropy variables as primary unknowns [43, 46], staggered grid methods [56, 68], and DG difference methods [98]. Here, we apply entropystable Gauss methods [17] available in Trixi.jl [78, 85] with efficient implementations [77]. While these schemes are more complex than collocation methods due to the entropy projection, they also possess favorable robustness properties for multidimensional simulations of underresolved compressible flows with variable density and smallscale features [18].
We consider a KelvinHelmholtz instability of the 2D compressible Euler equations of an ideal gas with ratio of specific heats \(\gamma = 1.4\) as in [18]. The domain \([1, 1]^2\) is equipped with periodic boundary conditions and the simulation is initialized with
where A is the Atwood number parameterizing the density contrast and
is a smoothed step function. We apply Gauss methods with 32 elements per coordinate direction and polynomials of degree \(p= 3\). We choose the entropyconservative flux of [75, 80, 81] for the volume terms and a local LaxFriedrichs flux at surfaces. We apply several representative RK methods to evolve the resulting entropydissipative semidiscretizations in time \(t \in [0, 5]\).
6.1 Mild Initial Conditions with Low Atwood Number
Table 4 shows a summary of the performance statistics of some representative RK methods for a low Atwood number \(A = 3/7\). The CFL number \(\nu\) was maximized manually up to three significant digits (so that the simulations did not crash). We used time integration methods with controllers recommended in [74] and a default tolerance \(\texttt {tol}= 10^{4}\); we set \(\texttt {tol}= 10^{5}\) for \(\textrm{RDPK}{4}(3){9}_\textrm{F}[3{\textrm{S}^{*}}_+]\), since the simulation crashed for the looser default tolerance.
The total number of function evaluations and time steps is always smaller with the errorbased step size control in this example, usually around 5%. In addition, the manual optimization of the CFL factor \(\nu\) required several simulation runs. Note that the optimal CFL factors are different from the ones used for the OrzsagTang vortex while the same tolerances were used. For this setup, \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\) is the most efficient method, followed by \(\textrm{RDPK}{3}(2){5}_\textrm{F}[3{\textrm{S}^{*}}_+]\).
6.2 Demanding Initial Conditions with High Atwood Number
Increasing the density contrast makes this underresolved test case more demanding [18]. Violations of positivity are more likely to occur and the sensitivity of the entropy variables for nearvacuum states can cause issues [98, 15]. In particular, the basic variables evolved in time for the Gauss methods violate positivity properties of the density/pressure around \(t \approx 3\) even for strict CFL factors such as \(\nu = 0.05\) for \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\), so that a standard CFLbased step size control is not possible. Similar problems occur for other RK methods.
In contrast, the errorbased step size control works directly in most cases, as shown in Table 5. We only needed to use a stricter tolerance \(\texttt {tol}= 10^{6}\) for \(\textrm{RDPK}{4}(3){9}_\textrm{F}[3{\textrm{S}^{*}}_+]\). This RK method has more stages than the other methods, making it more difficult to adapt quickly to changing conditions. Again, \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\) is the most efficient method, followed by \(\textrm{RDPK}{3}(2){5}_\textrm{F}[3{\textrm{S}^{*}}_+]\).
7 New Systems: Convenient Exploratory Research
A key feature of errorbased time stepping methods is that they require no knowledge of the maximum eigenvalues to select a stable explicit time step. This feature is particularly convenient when one wants to apply the existing numerical methods to new equation systems. To illustrate this, we consider the shallow water equations augmented with an Exner equation to account for interactions between the bottom topography and the fluid flow. The coupled system of equations in two spatial dimensions is
where h(x, y, t) is the water height, \(\pmb {v}(x,y,t)=(v_1(x,y,t),v_2(x,y,t))^\text{T}\) is the velocity field, b(x, y, t) is the evolving bottom topography, and g is the gravitational constant.
The last line in (18) is the Exner equation for the time evolution of the bottom topography b. It contains the constant \(\xi = 1/(1\sigma )\) where \(\sigma \in (0,1)\) is the porosity of the bottom material, and the solid transport discharge \(\pmb {q}=(q_1,q_2)^\text{T}\) as a function of the flow velocity \(\pmb {v}\). A closure model is required to describe how the bottom topography couples to the flow and defines the form of the \(\pmb {q}\) terms. This closure depends on the characteristics of the bottom material and flow, e.g., the grain size or Froude number. There exist many closure models for the solid transport discharge proposed and studied (analytically as well as numerically) in the literature; see e.g., [10, 30, 32, 64]. For the present study, we consider the simple and frequently used model due to Grass [32] that models the instantaneous sediment transport as a power law on the velocity field
In (19), the constant \(A_g\in [0,1]\) accounts for the porosity of the bottom sediment layer as well as the effects of the grain size and is usually determined from experimental data. When \(A_g\) is zero, there is no sediment transport and (18) reduces to the standard twodimensional shallow water equations. The interaction between the bottom and the water flow is weak when \(A_g\) is small and strong as \(A_g\) approaches one. The factor \(m\in [1,4]\) in the Grass model may also be determined from data; however, if one considers odd integer values of m, then (19) can be differentiated and the model remains valid for all values of the velocity field \(\pmb {v}\). For the remainder of this discussion, we will take \(m=3\) and the solid transport discharge will take the form
For the particular closure model (20), we investigate the flux Jacobian of the governing system of equations (18). This is because the choice of a stable explicit time step under a CFL condition requires knowledge of the fastest wave speed of the underlying system. For this, we compute the flux Jacobian in the xdirection, which incorporates nonconservative terms from the righthand side of (18), to be
Note that the analysis and results in the ydirection are analogous, so we will only consider (21) for this eigenvalue analysis for time step purposes. With the chosen Grass closure model, it has been shown that all the eigenvalues of (21) are real and the governing system (18) is hyperbolic [19, 26]. The eigenvalues of the Jacobian in the xdirection are \(v_1\) and the three roots of the characteristic polynomial
It is possible, although unwieldy, to apply Cardano’s formula to determine the real, distinct roots of (22). However, an accurate estimate to the maximum wave speed from (22) is necessary for the traditional CFLbased time stepping methods.
For many applications of interest, the characteristic flow speed of the water is much faster than the speed of the bottom topography [19, 45, 92]. In particular, for subcritical flow, the value of \(v_1\) is relatively small, so it is possible to use a loose bound of the flow speeds with \(v_1 \pm \sqrt{gh}\) where \(\sqrt{gh}\) is the surface wave speed. However, this loose bound applied to a CFL time step condition can result in excessive numerical diffusion that acts on the computed (slowly moving) bottom topography.
To approximate the solution of the shallow water equations with an Exner model (18), we use a DG spectral element method in space to create the semidiscretization and apply explicit time integration; see [95] for details. To couple the moving bottom topography b(x, y, t), we use an unsteady approach where the water flow and riverbed are calculated simultaneously. That is, the water flow can be either steady or unsteady and the changes in the bed update are considered to be significant, i.e., the wave speed of the bedupdating equation is of a similar magnitude to the wave speeds of the water flow. For this approach, the system of flow and bottom evolution equations are discretized simultaneously. This highorder DG method applied to the shallow water equations is well balanced for static and subcritical moving water flow [95, 96].
As an exemplary test case, we consider the evolution of a smooth sand hill bottom topography in a uniform flow governed by the shallow water equations. This test case is a twodimensional numerical simulation that was first proposed and analyzed by de Vriend [24]. It has become a common test case for benchmarking purposes in the moving bottom topography literature, e.g., [5, 19, 26, 45, 92], as it is easy to set up and demonstrates the ability of a numerical scheme to capture the morphology of a bottom topography governed with (18).
It involves the evolution of a conical sand dune in a channel with a nonerodible bottom on a square domain with dimensions 1 000 m \(\times\) 1 000 m. The initial conditions of the flow variables for this test case given in conservative variables are
and the initial form of the sediment layer is given by
We set the gravitational constant to be \(g = {9.8}\,{\textrm{m}/\textrm{s}^2}\). The porosity of the sediment material is \(\sigma = 0.4\) and the free parameter from the Grass model (20) is \(A_g = 0.001\). The parameter \(A_g\) corresponds to the coupling strength and speed of interaction between the sediment and the water flow [5, 45]. Taking a small value for \(A_g\) models, weak coupling and a simulation must be run for a longer time period to observe significant variations in the sediment layer. Therefore, we take the final time of this simulation to be \(t = 100\) h (360 000 s). The discharge value \(hv_1= {10}\;{\textrm{m}^2/\textrm{s}}\) from (23) gives the Froude number
Thus, the flow is in the subcritical regime and we must set boundary conditions accordingly; see, e.g., [65]. For this test case, the boundary conditions are taken to be walls on the top and bottom of the domain, the left is a subcritical inflow, and the right is a subcritical outflow. The complete setup can be found in the reproducibility repository [79].
For the spatial discretization, we divide the domain \(\varOmega = [0,1\,000]^2\) into 900 Cartesian elements. In each element, we use polynomials of degree \(p=5\) in each spatial direction for the DG approximation. This results in 32 400 DOFs for each equation in (18). The classic shallow water equations for water height, momentum, and bottom topography terms are discretized with the wellbalanced, highorder DG method described by Wintermeyer et al. [95]. The additional Exner equation for sediment transport is discretized with a standard DG spectral element approximation.
We first investigate the use of the errorbased time step control for this shallow water with sediment transport problem setup. We recorded the time step sizes and effective CFL numbers after every accepted time step for three representation RK methods. The tolerance is set to \(\texttt {tol}= 10^{7}\) for all errorbased time stepping methods considered. All three time integration techniques work directly for this test case. The behavior of the time step size and the effective CFL number are presented in Fig. 16. Initially, the time step size of the three methods increases before the time step settles to a stable value as the physical problem becomes a quasisteady flow where the sediment slowly evolves in the presence of the background flow.
We further investigate the effectiveness of the three time integrators with the errorbased step size control for this test case, as shown in Table 6. For this quasisteadystate configuration, we found that a stricter tolerance \(\texttt {tol}= 10^{7}\) for all the integration techniques tested performed the best in the sense of requiring a lower number of time steps. Here, we find that \(\textrm{RDPK}{4}(3){9}_\textrm{F}[3{\textrm{S}^{*}}_+]\) is the most efficient method, followed by \(\textrm{RDPK}{3}(2){5}_\textrm{F}[3{\textrm{S}^{*}}_+]\).
We examine the expected solution behavior as well as simulation results for the sediment transport problem under consideration. As time evolves, the conical dune (24) gradually spreads into a starshaped pattern. That is, the sediment will redistribute itself along a triangular shape with a particular spreading angle \(\alpha\) and the bulk shape of the dune will move with a speed \(c_0\) as it spreads [24, 45]. This process is illustrated in Fig. 17.
When the Grass model is used and the interaction between the flow and sediment is weak, that is \(A_g<0.01\), de Vriend [24] derived an analytical approximation for this spreading angle. In particular, when \(m=3\) in the Grass model (20), this spreading angle of a conical dune is
Next, we present simulation results obtained using \(\textrm{RDPK}{4}(3){9}_\textrm{F}[3{\textrm{S}^{*}}_+]\), which was found to be the most efficient. The numerical results from the other two time integration techniques were similar and will not be presented. Figure 18 shows the bottom topography at the initial and final times. From this figure, we see that the discretization captures the expected starshaped pattern of the sediment evolution.
We also estimate the spreading angle of a conical sand dune obtained for the numerical simulation [92]. This is done by examining the angle up to the isolevel 0.012 5 over time, as shown in Fig. 19. We are able to numerically estimate the spreading angle to be \(\alpha _{\text {num}} = {20.4}{^\circ }\). The numerical spreading angle of the current method compares well to the theoretical angle (26) as well as to similar angle studies done in the literature [5, 26, 45, 92].
We applied errorbased time stepping methods to the shallow water equations augmented with an Exner equation to model sediment transport. This served to demonstrate that it is convenient and straightforward to extend the existing code for a new system of equations and obtain a first set of numerical results. This was particularly useful for the considered shallow water Exner equations, because eigenvalue estimates for a general problem setup are unwieldy to obtain [19, 26]. These preliminary results for the evolution and spreading of a conical sand dune compared well to the theoretical solution behavior as well as results from the literature. The development of highorder DG methods for sediment transport problems is the subject of ongoing research. Details about the physical problem setup and numerical parameters as well as all code necessary to reproduce the figures in this section are available in the reproducibility repository for this article [79].
8 Summary and Conclusions
We have compared CFL and errorbased step size controls of explicit RK methods applied to DG semidiscretizations of compressible CFD problems. Our numerical experiments demonstrate that the errorbased step size control is convenient and robust in a wide range of applications, including industrially relevant complex geometries with curved meshes in multiple dimensions, shock capturing schemes, initial transient periods (cold start of a simulation), semidiscretizations involving a change of variables, and for exploratory research involving new systems of equations.
There are cures for some problems of the CFLbased step size control discussed in this article. For example, a cold start requiring an initially smaller CFL factor could run a few hundred time steps with a small CFL factor and use a larger CFL factor after a restart—or increase the CFL factor based on a usersupplied function in the first time steps. A change of linear stability behavior of shock capturing schemes could be handled by including the shock capturing indicator in the CFLbased step size control. However, this requires sophisticated additional mechanisms as implemented in some inhouse codes. On the other hand, the errorbased step size control worked right out of the box without further modifications for the problems considered here. It does not require special handling of many edge cases and comes with a reduced sensitivity with respect to usersupplied parameters.
While we have shown that the errorbased step size control is convenient and robust in many applications, including industrially relevant largescale computations of compressible CFD problems, it does not come with hard mathematical guarantees. For example, some invariant domain preserving methods come with some robustness properties guaranteed under certain CFL constraints; the errorbased step size control alone can usually not be proven to satisfy these constraints in all cases. Nevertheless, based on our experience, it is usually robust for suitably chosen shock capturing mechanisms.
As a byproduct of this research, we also compared several time integration methods for hyperbolic conservation laws and compressible flow problems. For lower Mach number flows or problems with positivity issues and/or strong shock capturing requirements, \({\textrm{SSP}}3(2)4[3{\textrm{S}^{*}}_{+}]\) is usually the most efficient scheme. Otherwise, \(\textrm{RDPK}{3}(2){5}_\textrm{F}[3\textrm{S}^{*}_+]\) and \(\textrm{RDPK}{4}(3){9}_\textrm{F}[3{\textrm{S}^{*}}_+]\) are usually the most efficient methods, where \(\textrm{RDPK}{3}(2){5}_\textrm{F}[3{\textrm{S}^{*}}_+]\) tends to be more robust and \(\textrm{RDPK}{4}(3){9}_\textrm{F}[3{\textrm{S}^{*}}_+]\) tends to be more efficient for stricter tolerances. Nevertheless, \(\textrm{BS}{3}(2){3}_\textrm{F}\) is still one of the best generalpurpose ODE solvers for problems like the ones considered in this article.
Future research includes the investigation of the errorbased step size control with adaptive mesh refinement and stabilization techniques such as positivitypreserving limiters [99]. These are usually implemented as callbacks outside of the inner time integration loop and are thus “not seen” by error estimators. Moreover, we would like to optimize new RK methods including error estimators and step size controllers for compressible CFD problems in other ranges than [2], e.g., focusing on lower Mach flows and dissipative effects.
Data Availability
The source code and datasets generated and analyzed for this study using open source codes are available in [79].
Notes
We use a superscript 0 to denote the initial data.
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Funding
Open Access funding enabled and organized by Projekt DEAL. Andrew Winters was funded through Vetenskapsrådet, Sweden Grant Agreement 202003642 VR. Some computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at Tetralith, partially funded by the Swedish Research Council under Grant Agreement No. 201805973. Hugo Guillermo Castro was funded through the award P20210004 of King Abdullah University of Science and Technology. Some of the simulations were enabled by the Supercomputing Laboratory and the Extreme Computing Research Center at King Abdullah University of Science and Technology. Gregor Gassner acknowledges funding through the KlausTschira Stiftung via the project “HiFiLab”. Gregor Gassner and Michael SchlottkeLakemper acknowledge funding from the Deutsche Forschungsgemeinschaft through the research unit “SNuBIC” (DFGFOR5409).
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Ranocha, H., Winters, A.R., Castro, H.G. et al. On ErrorBased Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics. Commun. Appl. Math. Comput. (2023). https://doi.org/10.1007/s4296702300264y
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DOI: https://doi.org/10.1007/s4296702300264y
Keywords
 Explicit RungeKutta (RK) methods
 Step size control
 Compressible fluid dynamics
 Adaptivity in space and time
 Shock capturing