High-order Spectral Difference Method Research Articles

Efficient dimension-by-dimension adaptive TENO-based limiter and troubled-cell indicator for nodal-based high-order spectral difference method

A new targeted essentially non-oscillatory (TENO) limiter with adaptive dissipation has been developed for the 3rd- and 4th-order spectral difference method. This limiter can potentially be useful for other nodal-based discontinuous high-order methods as well. Unlike traditional WENO limiters for these methods which limit low-order moments one by one, the new limiter is based on a direct convex combination of reconstruction polynomial candidates. This strategy ensures the good computational efficiency for high order reconstructions. The reconstruction stencils only involve nodal values from the target cell and its neighbors sharing faces with it. The reconstruction employs Hermite polynomials, making the new limiter compact. It can be implemented in a dimension-by-dimension manner for multi-dimensional problems, which is consistent with the spectral difference method. To further enhance efficiency, a TENO-based troubled-cell indicator is also developed to only activate limiters in troubled cells. Extensive one-dimensional and two-dimensional numerical experiments validate the performance of the new TENO-based limiter and indicator. In summary, the limiter is much less dissipative than WENO limiters and can resolve richer small-scale flow structures on coarse meshes. The indicator precisely marks out troubled cells, slightly improving over the KXRCF indicator.

GPU Implementation and Optimization of a High-Order Spectral Difference Method for Aeroacoustic Problems

GPU Implementation and Optimization of a High-Order Spectral Difference Method for Aeroacoustic Problems

Arbitrarily high-order accurate simulations of compressible rotationally constrained convection using a transfinite mapping on cubed-sphere grids

We present two major improvements over the Compressible High-ORder Unstructured Spectral difference (CHORUS) code published in Wang et al., “A compressible high-order unstructured spectral difference code for stratified convection in rotating spherical shells,” J. Comput. Phys. 290, 90–111 (2015). The new code is named CHORUS++ in this paper. Subsequently, we perform a series of efficient simulations for rotationally constrained convection (RCC) in spherical shells. The first improvement lies in the integration of the high-order spectral difference method with a boundary-conforming transfinite mapping on cubed-sphere grids, thus ensuring exact geometric representations of spherical surfaces on arbitrary sparse grids. The second improvement is on the adoption of higher-order elements (sixth-order) in CHORUS++ vs third-order elements for the original CHORUS code. CHORUS++ enables high-fidelity RCC simulations using sixth-order elements on very coarse grids. To test the accuracy and efficiency of using elements of different orders, CHORUS++ is applied to a laminar solar benchmark, which is characterized by columnar banana-shaped convective cells. By fixing the total number of solution degrees of freedom, the computational cost per time step remains unchanged. Nevertheless, using higher-order elements in CHORUS++ resolves components of the radial energy flux much better than using third-order elements. To obtain converged predictions, using sixth-order elements is 8.7 times faster than using third-order elements. This significant speedup allows global-scale fully compressible RCC simulations to reach equilibration of the energy fluxes on a small cluster of just 40 cores. In contrast, CHORUS simulations were performed by Wang et al. on supercomputers using approximately 10 000 cores. Using sixth-order elements in CHORUS++, we further carry out global-scale solar convection simulations with decreased rotational velocities. Interconnected networks of downflow lanes emerge and surround broader and weaker regions of upflow fields. A strong inward kinetic energy flux compensated by an enhanced outward enthalpy flux appears. These observations are all consistent with those published in the literature. Furthermore, CHORUS++ can be extended to magnetohydrodynamic simulations with potential applications to the hydromagnetic dynamo processes in the interiors of stars and planets.

An Arbitrarily High-order Spectral Difference Method with Divergence Cleaning (SDDC) for Compressible Magnetohydrodynamic Simulations on Unstructured Grids

This paper reports a recent development of the high-order spectral difference method with divergence cleaning (SDDC) for accurate simulations of both ideal and resistive magnetohydrodynamics (MHD) on curved unstructured grids consisting of high-order isoparametric quadrilateral elements. The divergence cleaning approach is based on the improved generalized Lagrange multiplier, which is thermodynamically consistent. The SDDC method can achieve an arbitrarily high order of accuracy in spatial discretization, as demonstrated in the test problems with smooth solutions. The high-order SDDC method combined with the artificial dissipation method can sharply capture shock interfaces with the oscillation-free property and resolve small-scale vortex structures and density fluctuations on relatively sparse grids. The robustness of the codes is demonstrated through long time simulations of ideal MHD problems with progressively interacting shock structures, resistive MHD problems with high Lundquist numbers, and viscous resistive MHD problems on complex curved domains.

A Divergence-Free High-Order Spectral Difference Method with Constrained Transport for Ideal Compressible Magnetohydrodynamics

When the high-order Spectral Difference (SD) method is used to discretize ideal magnetohydrodynamic (MHD) equations, it is challenging to satisfy the divergence-free constraint for the magnetic field over long time integration. To ensure that the discrete equals to zero exactly and globally, the SD method is integrated with an unstaggered Constrained Transport approach (SDCT) by replacing the magnetic field with the curl of the magnetic potential at every time step. The SDCT method stores the variables for the hydrodynamics and the magnetic field at the same set of solution points, which avoids designing 2D Riemann solvers and preserves the compactness of the stencil for spatial discretization. Moreover, the additional computational cost is less than 1/8 of that without the constrained transport. Meanwhile, the SDCT method is found to have excellent convergence in test cases with and without shocks.

An arbitrary high-order Spectral Difference method for the induction equation

We study in this paper three variants of the high-order Discontinuous Galerkin (DG) method with Runge-Kutta (RK) time integration for the induction equation, analysing their ability to preserve the divergence-free constraint of the magnetic field. To quantify divergence errors, we use a norm based on both a surface term, measuring global divergence errors, and a volume term, measuring local divergence errors. This leads us to design a new, arbitrary high-order numerical scheme for the induction equation in multiple space dimensions, based on a modification of the Spectral Difference (SD) method [1] with ADER time integration [2]. It appears as a natural extension of the Constrained Transport (CT) method. We show that it preserves ∇⋅B→=0 exactly by construction, both in a local and a global sense. We compare our new method to the 3 RKDG variants and show that the magnetic energy evolution and the solution maps of our new SD-ADER scheme are qualitatively similar to the RKDG variant with divergence cleaning, but without the need for an additional equation and an extra variable to control the divergence errors.

Entropy preserving low dissipative shock capturing with wave-characteristic based sensor for high-order methods

Shock capturing procedures are required to stabilise numerical simulations of gas dynamics problems featuring non-isentropic discontinuities. In the present work, particular attention is focused on the expected non-monotonicity of the entropy profile across shock waves. A peculiar physical property which was not considered so far in the evaluation of shock capturing techniques. In the context of high-order spectral difference methods and using most recent discontinuity sensors based on the decay rate of the modes of the amplitude of characteristic waves, results show how the choice of a physical-based procedure (additional viscosity) returns a better description of shocks compared to approaches relying on the direct addition of a Laplacian term in the solved equations. Various canonical compressible flows are simulated, in one-, two-, and three-dimensional setups, to illustrate the performance and flexibility of the proposed approach. It is shown that the addition of a well-calibrated bulk viscosity is capable of smoothing out discontinuities without an excessive damping of vortical structures, preserving also specific compressible flow physics, as the non-monotonic entropy profiles through the shocks.

Impact of Number Representation for High-Order Implicit Large-Eddy Simulations

High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. Over the past few years, they have shown promise in enabling implicit large-eddy simulations within the vicinity of complex geometrical configurations. However, the cost of such simulations remains prohibitive. In this paper, the impact of number representation on the efficiency and efficacy of these methods is considered. Theoretical performance models are derived and assessed. Specifically, four test cases are considered: 1) the viscous Taylor–Green vortex, 2) flow over a circular cylinder, 3) flow through a T106c low-pressure turbine cascade, and 4) flow around a NACA 0021 in deep stall. It is found that the use of single (in lieu of double) precision arithmetic does not have a significant impact on accuracy. The performance of the resulting simulations was found to improve between 1.4 and 2.9 times.

A new high-order spectral difference method for simulating viscous flows on unstructured grids with mixed-element meshes

In the present study, a spectral difference (SD) method is developed for viscous flows on meshes with a mixture of triangular and quadrilateral elements. The standard SD method for triangular elements, which employs Lagrangian interpolating functions for fluxes, is not stable when the designed accuracy of spatial discretization is third order or higher. Unlike the standard SD method, the method examined here uses vector interpolating functions in the Raviart-Thomas (RT) spaces to construct continuous flux functions on reference elements. The spectral-difference Raviart-Thomas (SDRT) method was originally proposed by Balan et al. [1] and implemented on triangular-element meshes for invisid flow only. Our present results demonstrated that the SDRT method is stable and high-order accurate in two dimensions (2D) for a number of test problems by using triangular-, quadrilateral-, and mixed-element meshes for both inviscid and viscous flows. A stability analysis is also performed for mixed elements. We found our current SDRT scheme is stable for fourth-order accurate spatial discretizations. However, a stable fifth-order SDRT scheme remains to be identified.

High order spectral difference lattice Boltzmann method for incompressible hydrodynamics

This work presents a lattice Boltzmann equation (LBE) based high order spectral difference method for incompressible flows. In the present method, the spectral difference (SD) method is adopted to discretize the convection and collision term of the LBE to obtain high order (≥3) accuracy. Because the SD scheme represents the solution as cell local polynomials and the solution polynomials have good tensor-product property, the present spectral difference lattice Boltzmann method (SD-LBM) can be implemented on arbitrary unstructured quadrilateral meshes for effective and efficient treatment of complex geometries. Thanks to only first oder PDEs involved in the LBE, no special techniques, such as hybridizable discontinuous Galerkin method (HDG), local discontinuous Galerkin method (LDG) and so on, are needed to discrete diffusion term, and thus, it simplifies the algorithm and implementation of the high order spectral difference method for simulating viscous flows. The proposed SD-LBM is validated with four incompressible flow benchmarks in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the lid-driven cavity flow without singularity at the two top corners–Burggraf flow; and (c) the unsteady Taylor–Green vortex flow; (d) the Blasius boundary-layer flow past a flat plate. Computational results are compared with analytical solutions of these cases and convergence studies of these cases are also given. The designed accuracy of the proposed SD-LBM is clearly verified.

Implementation of spectral difference method on overset grids for compressible inviscid flows

This work investigates the use of a high-order spectral difference (SD) method on overset grids for the solution of the Euler equations. A spectral difference method of up to fourth order for hexahedral elements is employed as the spatial discretization scheme and the explicit four-stage Runge–Kutta method is utilized for time integration. In order to preserve the high-order accuracy throughout the whole computation domain, an interpolation method for overlapping interfaces is developed. To validate this interpolation method and the accuracy of the overall algorithm, the inviscid convection of a vortex and subsonic flow over a wing are calculated. Then the flow over a hovering rotor is simulated by the high-order SD method on overset grids. The numerical results demonstrated the capability and accuracy of the present algorithm on overset grids.

Detached eddy simulation of a highlift airfoil noise with spectral difference method

In this study the noise generated by the 30P30N highlift airfoil is simulated with a high order computational aeroacoustics code. To handle the complex geometry of the airfoil, the high order spectral difference method based on unstructured mesh is used for spatial discretization. The multi-time-step method based on Adam-Bashforth scheme is utilized for time marching, and a speed-up ratio greater than 10 is obtained comparing with the single time step method. The detached eddy simulation method is conducted for turbulence modeling. Radiation and outflow non-reflective boundary conditions are used in the far field and downstream. In this simulation, the inflow Mach number is 0.17, and the Reynolds number based on the inflow velocity and the chord length of the airfoil is 1.7e6. Three attack angles, which are 4, 5.5, and 8.5 degrees, respectively, are considered in this study. The dynamic pressure on the surface of the slat is sampled and compared with the experimental data by other researchers, and a good agreement is obtained. The far field noise, computed with the permeable Ffowcs Williams-Hawkings (FW-H) integration method, is analyzed and compared with the experimental data. The noise source is analyzed and identified with a correlation analysis of the dynamic pressure on the slat and the near field noise.

Graphics-Processing-Unit-Accelerated Multiphase Computational Tool for Asteroid Fragmentation/Pulverization Simulation

The simulation of asteroid target fragmentation or pulverization is a challenging task that demands efficient and accurate numerical methods with large computational power. To this end, the high-order spectral difference method is implemented with graphics-processing-unit computing. Hypervelocity kinetic-energy impactors are of practical interest, which generate high-pressure deformational shock waves in the target bodies upon impact. Due to the extremely short deformation time associated with hypervelocity impact, the material behaves in a similar manner to a compressible fluid, and the compressible Euler equations can be applied. To model the multiple material interactions, an -phase equation model is adopted into the spectral difference method. All simulations presented are solved with graphics processing units, producing solutions orders of magnitude faster than the central-processing-unit counterpart. Several impact cases are compared, including a heavy impactor and multiple impactor system, against an asteroid target. Orbital dispersion effectiveness is evaluated, and results indicate that the multiple impactor system outperforms the single heavy impactor.

GPU-accelerated computational tool for studying the effectiveness of asteroid disruption techniques

This paper presents the development of a new Graphics Processing Unit (GPU) accelerated computational tool for asteroid disruption techniques. Numerical simulations are completed using the high-order spectral difference (SD) method. Due to the compact nature of the SD method, it is well suited for implementation with the GPU architecture, hence solutions are generated at orders of magnitude faster than the Central Processing Unit (CPU) counterpart. A multiphase model integrated with the SD method is introduced, and several asteroid disruption simulations are conducted, including kinetic-energy impactors, multi-kinetic energy impactor systems, and nuclear options. Results illustrate the benefits of using multi-kinetic energy impactor systems when compared to a single impactor system. In addition, the effectiveness of nuclear options is observed.

A compressible high-order unstructured spectral difference code for stratified convection in rotating spherical shells

We present a novel and powerful Compressible High-ORder Unstructured Spectral-difference (CHORUS) code for simulating thermal convection and related fluid dynamics in the interiors of stars and planets. The computational geometries are treated as rotating spherical shells filled with stratified gas. The hydrodynamic equations are discretized by a robust and efficient high-order Spectral Difference Method (SDM) on unstructured meshes. The computational stencil of the spectral difference method is compact and advantageous for parallel processing. CHORUS demonstrates excellent parallel performance for all test cases reported in this paper, scaling up to 12 000 cores on the Yellowstone High-Performance Computing cluster at NCAR. The code is verified by defining two benchmark cases for global convection in Jupiter and the Sun. CHORUS results are compared with results from the ASH code and good agreement is found. The CHORUS code creates new opportunities for simulating such varied phenomena as multi-scale solar convection, core convection, and convection in rapidly-rotating, oblate stars.

PyFR: An open source framework for solving advection–diffusion type problems on streaming architectures using the flux reconstruction approach

High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. The Flux Reconstruction (FR) approach unifies various high-order schemes for unstructured grids within a single framework. Additionally, the FR approach exhibits a significant degree of element locality, and is thus able to run efficiently on modern streaming architectures, such as Graphical Processing Units (GPUs). The aforementioned properties of FR mean it offers a promising route to performing affordable, and hence industrially relevant, scale-resolving simulations of hitherto intractable unsteady flows within the vicinity of real-world engineering geometries. In this paper we present PyFR, an open-source Python based framework for solving advection–diffusion type problems on streaming architectures using the FR approach. The framework is designed to solve a range of governing systems on mixed unstructured grids containing various element types. It is also designed to target a range of hardware platforms via use of an in-built domain specific language based on the Mako templating engine. The current release of PyFR is able to solve the compressible Euler and Navier–Stokes equations on grids of quadrilateral and triangular elements in two dimensions, and hexahedral elements in three dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented for various benchmark flow problems, single-node performance is discussed, and scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The software is freely available under a 3-Clause New Style BSD license (see www.pyfr.org). Program summaryProgram title: PyFR v0.1.0Catalogue identifier: AETY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: New style BSD licenseNo. of lines in distributed program, including test data, etc.: 12733No. of bytes in distributed program, including test data, etc.: 214183Distribution format: tar.gzProgramming language: Python, CUDA and C.Computer: Variable, up to and including GPU clusters.Operating system: Recent version of Linux/UNIX.RAM: Variable, from hundreds of megabytes to gigabytes.Classification: 6.5, 12.External routines: Python 2.7, numpy, PyCUDA, mpi4py, SymPy, MakoNature of problem:Compressible Euler and Navier–Stokes equations of fluid dynamics; potential for any advection–diffusion type problem.Solution method:High-order flux reconstruction approach suitable for curved, mixed, unstructured grids.Unusual features:Code makes extensive use of symbolic manipulation and runtime code generation through a domain specific language.Running time:Many small problems can be solved on a recent workstation in minutes to hours.

Parallel spectral difference method for predicting 3D vortex-induced vibrations

This paper presents high-fidelity simulations of the vortex induced vibration (VIV) phenomena using a new computational model based on the high-order spectral difference (SD) method on 2D and 3D unstructured grids. The SD method constructs continuous fields within each cell with a Riemann solver to compute the inviscid fluxes at the cell interfaces and an averaging mechanism to compute the viscous fluxes. This method has shown promise in the past as a highly accurate, yet sufficiently fast method for solving unsteady viscous compressible flows on moving and deforming grids. A 4th-order, 5-stage Runge–Kutta scheme is used to advance time. In this viscous, compressible flow solver the displacement of an elastically mounted bluff body has been coupled to the lift force created by unsteady vortex shedding. The solver is validated against previously published numerical and experimental data for a single, elastically mounted cylinder. Preliminary studies into 2 cylinder wake galloping and determining the energy transfer coefficient have been performed and comparisons between 2D and 3D predictions have been made.

A stable high-order Spectral Difference method for hyperbolic conservation laws on triangular elements

Numerical schemes using piecewise polynomial approximation are very popular for high order discretization of conservation laws. While the most widely used numerical scheme under this paradigm appears to be the Discontinuous Galerkin method, the Spectral Difference scheme has often been found attractive as well, because of its simplicity of formulation and implementation. However, recently it has been shown that the scheme is not linearly stable on triangles. In this paper we present an alternate formulation of the scheme, featuring a new flux interpolation technique using Raviart–Thomas spaces, which proves stable under a similar linear analysis in which the standard scheme failed. We demonstrate viability of the concept by showing linear stability both in the semi-discrete sense and for time stepping schemes of the SSP Runge–Kutta type. Furthermore, we present convergence studies, as well as case studies in compressible flow simulation using the Euler equations.

VALIDATION AND APPLICATION OF AN HIGH-ORDER SPECTRAL DIFFERENCE METHOD FOR FLOW INDUCED NOISE SIMULATION

The main goal of this paper is to develop an efficient numerical algorithm to compute the radiated far field noise provided by an unsteady flow field from bodies in arbitrary motion. The method computes a turbulent flow field in the near fields using a high-order spectral difference method coupled with large-eddy simulation approach. The unsteady equations are solved by advancing in time using a second-order backward difference formulae scheme. The nonlinear algebraic system arising from the time discretization is solved with the nonlinear lower–upper symmetric Gauss–Seidel algorithm. In the second step, the method calculates the far field sound pressure based on the acoustic source information provided by the first step simulation. The method is based on the Ffowcs Williams–Hawkings approach, which provides noise contributions for monopole, dipole and quadrupole acoustic sources. This paper will focus on the validation and assessment of this hybrid approach using different test cases. The test cases used are: a laminar flow over a two-dimensional (2D) open cavity at Re = 1.5 × 103 and M = 0.15 and a laminar flow past a 2D square cylinder at Re = 200 and M = 0.5. In order to show the application of the numerical method in industrial cases and to assess its capability for sound field simulation, a three-dimensional turbulent flow in a muffler at Re = 4.665 × 104 and M = 0.05 has been chosen as a third test case. The flow results show good agreement with numerical and experimental reference solutions. Comparison of the computed noise results with those of reference solutions also shows that the numerical approach predicts noise accurately.

A high-order spectral difference method for unstructured dynamic grids

A high-order spectral difference (SD) method has been further extended to solve the three dimensional compressible Navier–Stokes (N–S) equations on deformable dynamic meshes. In the SD method, the solution is approximated with piece-wise continuous polynomials. The elements are coupled with common Riemann fluxes at element interfaces. The extension to deformable elements necessitates a time-dependent geometric transformation. The Geometric Conservation Law (GCL), which is introduced in the time-dependent transformation from the physical domain to the computational domain, has been discussed and implemented for both explicit and implicit time marching methods. Accuracy studies are performed with a vortex propagation problem, demonstrating that the spectral difference method can preserve high-order accuracy on deformable meshes. Further applications of the method to several moving boundary problems including bio-inspired flow problems are shown in the paper to demonstrate the capability of the developed method.