NodePy: A package for the analysis of numerical ODE solvers

Ketcheson, David I.; Ranocha, Hendrik; Parsani, Matteo; Waheed, Umair bin; Hadjimichael, Yiannis

doi:10.21105/joss.02515

“…We ran simulations of all three test cases described above and measured the performance of each scheme (in terms of the number of right-hand side evaluations). We used a brute-force search over the domain ∈ [10 −8 , 10 −1 ] , sampling at each power of ten, and 1 ∈ [0.1, 1.0] , 2 ∈ [−0.4, −0.05] , 3 ∈ [0.0, 0.1] sampling at an interval of 0.01 in each parameter, and restricting a priori to parameter values yielding step size control stability for the given scheme (computed using NodePy [43]). The final time for these simulations was set to t = 8 for ( 14), t = 4 for (15), and t = 20 for ( 16) to make the brute-force optimization feasible.…”

Section: Optimization Of Step Size Controllersmentioning

confidence: 99%

Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

Ranocha

¹

,

Dalcín

²

,

Parsani

³

et al. 2021

Commun. Appl. Math. Comput.

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We develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice. We compare a wide range of error-control-based methods, along with the common approach in which step size control is based on the Courant-Friedrichs-Lewy (CFL) number. The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances, while additionally providing control of the temporal error at tighter tolerances. The numerical examples include challenging industrial CFD applications.

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“…For this, we used the algorithms implemented in RK-Opt [61], which are based on MATLAB [62]. We used NodePy [63] to verify the desired properties of the resulting five-stage second-order method RK3S* with low-storage coefficients listed in Table 1. The minimum-storage implementation [60] is shown in Algorithm 1.…”

Section: Time Integration Schemesmentioning

confidence: 99%

A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics

Schlottke-Lakemper

¹

,

Winters

²

,

Ranocha

³

et al. 2021

Journal of Computational Physics

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One of the challenges when simulating astrophysical flows with self-gravity is to compute the gravitational forces. In contrast to the hyperbolic hydrodynamic equations, the gravity field is described by an elliptic Poisson equation. We present a purely hyperbolic approach by reformulating the elliptic problem into a hyperbolic diffusion problem, which is solved in pseudotime using the same explicit high-order discontinuous Galerkin method we use for the flow solution. The flow and the gravity solvers operate on a joint hierarchical Cartesian mesh and are two-way coupled via the source terms. A key benefit of our approach is that it allows the reuse of existing explicit hyperbolic solvers without modifications, while retaining their advanced features such as non-conforming and solution-adaptive grids. By updating the gravitational field in each Runge-Kutta stage of the hydrodynamics solver, high-order convergence is achieved even in coupled multi-physics simulations. After verifying the expected order of convergence for single-physics and multi-physics setups, we validate our approach by a simulation of the Jeans gravitational instability. Furthermore, we demonstrate the full capabilities of our numerical framework by computing a self-gravitating Sedov blast with shock capturing in the flow solver and adaptive mesh refinement for the entire coupled system.

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“…∈ [−0.4, −0.05], 3 ∈ [0.0, 0.1] sampling at an interval of 0.01 in each parameter, and restricting a priori to parameter values yielding step size control stability for the given scheme (computed using NodePy[43]). The final times for these simulations were set to = 8 for (5.3), = 4 for (5.4), and = 20 for (5.5) to make the brute-force optimization feasible.…”

mentioning

confidence: 99%

Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

Ranocha,

Dalcin,

Parsani

et al. 2021

Preprint

Self Cite

0

View full text Add to dashboard Cite

We develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice. We compare a wide range of error-control-based methods, along with the common approach in which step size control is based on the Courant-Friedrichs-Lewy (CFL) number. The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances, while additionally providing control of the temporal error at tighter tolerances. The numerical examples include challenging industrial CFD applications.

show abstract

NodePy: A package for the analysis of numerical ODE solvers

Cited by 6 publications

References 21 publications

Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics

Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

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