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

  title={Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics},
  author={Hendrik Ranocha and Lisandro Dalcin and Matteo Parsani and David I. Ketcheson},
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… 
ARKODE: A flexible IVP solver infrastructure for one-step methods
The role of ARKODE is presented within the SUNDIALS suite of time integration and nonlinear solver libraries, the core ARKode infrastructure for utilities common to large classes of one-step methods, as well as its use of “time stepper” modules enabling easy incorporation of novel algorithms into the library.
On the Entropy Projection and the Robustness of High Order Entropy Stable Discontinuous Galerkin Schemes for Under-Resolved Flows
It is demonstrated numerically that entropy stable Discontinuous Galerkin (DG) methods which incorporate an “entropy projection” are less likely to require additional limiting to retain positivity for certain types of flows.
Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing
An assessment of Julia for simulation-focused scientific computing, an area that is still dominated by traditional high-performance computing languages such as C, C++, and Fortran.
Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous Galerkin methods for conservation laws
This paper aims to demonstrate the efforts towards in-situ applicability of EMMARM, the objective of which is to provide real-time information about the response of the immune system to x-ray diffraction.


Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods
It is demonstrated how control theory can be used to analyze and improve the standard stepsize control algorithm and a dynamic model that takes this behavior into account is derived for explicit Runge-Kutta methods.
Explicit two‐step Runge‐Kutta methods for computational fluid dynamics solvers
A class of two‐step Runge‐Kutta (TSRK) methods of high order with low number of stages, for time discretization of differential systems resulting from space discretized of weakly compressible Navier‐Stokes equations are identified and tested.
On the robustness and performance of entropy stable discontinuous collocation methods for the compressible Navie-Stokes equations
It is numerically show that low and high order entropy stable discontinuous spatial discretizations based on summation-by-part operators and simultaneous-approximation-terms technique provides an essential step toward a truly enabling technology in terms of reliability and robustness for both under-resolved turbulent flow simulations and flows with discontinuities.
Relaxation Runge-Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations
The framework of inner product norm preserving relaxation Runge-Kutta methods is extended to general convex quantities and is proved analytically and demonstrated in several numerical examples, including applications to high-order entropy-conservative and entropy-stable semi-discretizations on unstructured grids for the compressible Euler and Navier-Stokes equations.