Low-storage Runge-Kutta Scheme Research Articles

OpenSBLI: Automated code-generation for heterogeneous computing architectures applied to compressible fluid dynamics on structured grids

OpenSBLI is an open-source code-generation system for compressible fluid dynamics (CFD) on heterogeneous computing architectures. Written in Python, OpenSBLI is an explicit high-order finite-difference solver on structured curvilinear meshes. Shock-capturing is performed by a choice of high-order Weighted Essentially Non-Oscillatory (WENO) or Targeted Essentially Non-Oscillatory (TENO) schemes. OpenSBLI generates a complete CFD solver in the Oxford Parallel Structured (OPS) domain specific language. The OPS library is embedded in C code, enabling massively-parallel execution of the code on a variety of high-performance-computing architectures, including GPUs. The present paper presents a code base that has been completely rewritten from the earlier proof of concept Jacobs et al. (2017) [7], allowing shock capturing, coordinate transformations for complex geometries, and a wide range of boundary conditions, including solid walls with and without heat transfer. A suite of validation and verification cases are presented, plus demonstration of a large-scale Direct Numerical Simulation (DNS) of a transitional Shockwave Boundary Layer Interaction (SBLI). The code is shown to have good weak and strong scaling on multi-GPU clusters. We demonstrate that code-generation and domain specific languages are suitable for performing efficient large-scale simulations of complex fluid flows on emerging computing architectures. Program summaryProgram Title: OpenSBLI code-generation framework for compressible fluid dynamics on heterogeneous architecturesCPC Library link to program files:https://doi.org/10.17632/3sdb6hck2c.1Developer's repository link:https://github.com/opensbli/opensbliLicensing provisions: GPLv3Programming languages: Python, C/C++, OPS DSLNature of problem: The compressible 3D Navier-Stokes equations are solved via Implicit Large Eddy Simulation (ILES) or Direct Numerical Simulation (DNS).Solution method: OpenSBLI [1,2] is a Python-based code-generation system that uses symbolic algebra to generate a complete CFD solver in C/C++. The basic algorithm is a stencil-based finite-difference solver on structured curvilinear meshes. Shock-capturing is performed by a selection of high-order Weighted/Targeted Essentially Non-Oscillatory (WENO/TENO) schemes. Explicit low-storage Runge-Kutta schemes are used for time-advancement.Additional comments including restrictions and unusual features: The generated code is compliant with the Oxford Parallel Structured (OPS) [3] software library. OpenSBLI/OPS executables can be generated for the OpenMP, MPI, CUDA, OpenCL, and OpenACC parallel programming paradigms. Multi-GPU support is available via combinations of MPI with CUDA, OpenCL or OpenACC.

A low—storage Runge—Kutta OpenFOAM solver for compressible low—Mach number flows: aeroacoustic and thermo—fluid dynamic applications

A solver for compressible Navier–Stokes equations is presented in this paper. Low-storage RungeKutta schemes were adopted for time integration; on the other hand the finite volume approach available within OpenFOAM library has been adopted for space discretization. Kurganov-Noelle-Petrova approach was used for convective terms, while central schemes for diffusive ones. The aforementioned techniques were selected and tested in order to allow the possibility of solving a broad range of physical phenomena with particular emphasis to aeroacoustic and thermo-fluid dynamic problems. Indeed, that standard OpenFOAM solution techniques produce an unacceptable dissipation for acoustic phenomena computations. Non–reflective boundary treatment was also considered to avoid spurious numerical reflections. The reliability and the robustness of the solver is proved by computing several benchmarks. Lastly, the impact of the thermal boundary conditions on the sound propagation was analyzed.

Petascale simulations of compressible flows with interfaces

We demonstrate a high throughput software for the efficient simulation of compressible multicomponent flow on high performance computing platforms. The discrete problem is represented on structured three-dimensional grids with non-uniform resolution. Discontinuous flow features are captured using a diffuse interface method. A distinguishing characteristic of the method is the proper treatment of the interface zone as a mixing region of liquid and gas. The governing equations are discretized by a Godunov-type finite volume method with explicit time stepping using a low-storage Runge-Kutta scheme. The presented flow solver Cubism-MPCF is based on our Cubism library which enables a highly optimized framework for the efficient treatment of stencil based problems on multicore architectures. The framework is general and not limited to applications in fluid dynamics. We validate our solver by classical benchmark examples. Furthermore, we examine a highly-resolved shock-induced bubble collapse and a cloud of O(103) collapsing bubbles, which demonstrate the high potential of the proposed framework and solver.

Dynamic earthquake rupture simulations on nonplanar faults embedded in 3D geometrically complex, heterogeneous elastic solids

Dynamic propagation of shear ruptures on a frictional interface in an elastic solid is a useful idealization of natural earthquakes. The conditions relating discontinuities in particle velocities across fault zones and tractions acting on the fault are often expressed as nonlinear friction laws. The corresponding initial boundary value problems are both numerically and computationally challenging. In addition, seismic waves generated by earthquake ruptures must be propagated for many wavelengths away from the fault. Therefore, reliable and efficient numerical simulations require both provably stable and high order accurate numerical methods.We present a high order accurate finite difference method for: a) enforcing nonlinear friction laws, in a consistent and provably stable manner, suitable for efficient explicit time integration; b) dynamic propagation of earthquake ruptures along nonplanar faults; and c) accurate propagation of seismic waves in heterogeneous media with free surface topography.We solve the first order form of the 3D elastic wave equation on a boundary-conforming curvilinear mesh, in terms of particle velocities and stresses that are collocated in space and time, using summation-by-parts (SBP) finite difference operators in space. Boundary and interface conditions are imposed weakly using penalties. By deriving semi-discrete energy estimates analogous to the continuous energy estimates we prove numerical stability. The finite difference stencils used in this paper are sixth order accurate in the interior and third order accurate close to the boundaries. However, the method is applicable to any spatial operator with a diagonal norm satisfying the SBP property. Time stepping is performed with a 4th order accurate explicit low storage Runge-Kutta scheme, thus yielding a globally fourth order accurate method in both space and time. We show numerical simulations on band limited self-similar fractal faults revealing the complexity of rupture dynamics on rough faults.

Low-storage implicit/explicit Runge–Kutta schemes for the simulation of stiff high-dimensional ODE systems

Implicit/explicit (IMEX) Runge–Kutta (RK) schemes are effective for time-marching ODE systems with both stiff and nonstiff terms on the RHS; such schemes implement an (often A-stable or better) implicit RK scheme for the stiff part of the ODE, which is often linear, and, simultaneously, a (more convenient) explicit RK scheme for the nonstiff part of the ODE, which is often nonlinear. Low-storage RK schemes are especially effective for time-marching high-dimensional ODE discretizations of PDE systems on modern (cache-based) computational hardware, in which memory management is often the most significant computational bottleneck. In this paper, we develop and characterize eight new low-storage implicit/explicit RK schemes which have higher accuracy and better stability properties than the only low-storage implicit/explicit RK scheme available previously, the venerable second-order Crank–Nicolson/Runge–Kutta–Wray (CN/RKW3) algorithm that has dominated the DNS/LES literature for the last 25 years, while requiring similar storage (two, three, or four registers of length N) and comparable floating-point operations per timestep.

Efficient low-storage Runge–Kutta schemes with optimized stability regions

A variety of numerical calculations, especially when considering wave propagation, are based on the method-of-lines, where time-dependent partial differential equations (PDEs) are first discretized in space. For the remaining time-integration, low-storage Runge–Kutta schemes are particularly popular due to their efficiency and their reduced memory requirements. In this work, we present a numerical approach to generate new low-storage Runge–Kutta (LSRK) schemes with optimized stability regions for advection-dominated problems. Adapted to the spectral shape of a given physical problem, those methods are found to yield significant performance improvements over previously known LSRK schemes. As a concrete example, we present time-domain calculations of Maxwell’s equations in fully three-dimensional systems, discretized by a discontinuous Galerkin approach.

An efficient discontinuous Galerkin method for aeroacoustic propagation

SUMMARYAn efficient discontinuous Galerkin formulation is applied to the solution of the linearized Euler equations and the acoustic perturbation equations for the simulation of aeroacoustic propagation in two‐dimensional and axisymmetric problems, with triangular and quadrilateral elements. To improve computational efficiency, a new strategy of variable interpolation order is proposed in addition to a quadrature‐free approach and parallel implementation. Moreover, an accurate wall boundary condition is formulated on the basis of the solution of the Riemann problem for a reflective wall. Time discretization is based on a low dissipation formulation of a fourth‐order, low storage Runge–Kutta scheme. Along the far‐field boundaries a perfectly matched layer boundary condition is used. For the far‐field computations, the integral formulation of Ffowcs Williams and Hawkings is coupled with the near‐field solver. The efficiency and accuracy of the proposed variable order formulation is assessed for realistic geometries, namely sound propagation around a high‐lift airfoil and the Munt problem. Copyright © 2011 John Wiley & Sons, Ltd.

Comparison of Low-Storage Runge-Kutta Schemes for Discontinuous Galerkin Time-Domain Simulations of Maxwell's Equations

Comparison of Low-Storage Runge-Kutta Schemes for Discontinuous Galerkin Time-Domain Simulations of Maxwell's Equations

A general strategy for the optimization of Runge–Kutta schemes for wave propagation phenomena

We analyze optimized explicit Runge–Kutta schemes (RK) for computational aeroacoustics, and wave propagation phenomena in general. Exploiting the analysis developed in [S. Pirozzoli, Performance analysis and optimization of finite-difference schemes for wave propagation problems, J. Comput. Phys. 222 (2007) 809–831], we rigorously evaluate the performance of several time integration schemes in terms of appropriate error and cost metrics, and provide a general strategy to design Runge–Kutta methods tailored for specific applications. We present families of optimized second- and third-order Runge–Kutta schemes with up to seven stages, and describe their implementation in the framework of Williamson’s 2 N -storage formulation [J.H. Williamson, Low-storage Runge–Kutta schemes, J. Comput. Phys. 35 (1980) 48–56]. Numerical simulations of the 1D linear advection equation and of the 2D linearized Euler equations are performed to demonstrate the validity of the theory and to quantify the improvement provided by optimized schemes.

HIGHER-ORDER COMPACT SCHEMES FOR NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS, PART I: THEORETICAL DEVELOPMENT

A higher-order-accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for fluid flow problems. It is based on low-storage Runge-Kutta schemes for temporal discretization and fourth- and sixth-order compact finite-difference schemes for spatial discretization. New insights are presented on the elimination of the odd-even decoupling problem in the solution of the pressure Poisson equation. For consistent global accuracy, it is necessary to employ the same order of accuracy in the discretization of the Poisson equation. Accuracy and robustness issues are addressed by application to several pertinent benchmark problems in Part II.

Direct simulation of compressible turbulence in a shear flow

The purpose of this study is to investigate compressibility effects on the turbulence in homogeneous shear flow. Physical experiments have not been and perhaps cannot be performed for homogeneous shear flows at flow speeds which are sufficiently high to introduce compressible effects on the turbulence. However, direct numerical simulation (DNS) of this problem could provide meaningful data, especially since DNS for the incompressible problem ([1], [2]) has been successful in giving realistic flow fields akin to those obtained in physical experiments. The present simulations of the homogeneous shear problem augment and extend our previous results ([3], [4]) for compressible isotropic turbulence. The compressible Navier-Stokes equations are written in a frame of reference moving with the mean flow u. We use a Fourier collocation method for the spatial discretization of the governing equations. A third order, low storage Runge-Kutta scheme is used for advancing the solution in time. Up to 1283 points are used for the simulations. The simulated turbulent fields have Taylor microscale Reynolds numbers Re? up to 35 and turbulent Mach numbers Mt up to 0.6. The results are summarized below. We find that an increase in initial compressibility level, either due to increased turbulent Mach number or increased dilatational fraction of the velocity field decreases the growth of turbulent kinetic energy (and all its components) in the case of homogeneous shear flow. This is reminiscent of the reduction in turbulent velocity intensities observed in experiments on supersonic free shear