Matteo Parsani

PYCLAW: ACCESSIBLE, EXTENSIBLE, SCALABLE TOOLS FOR WAVE PROPAGATION PROBLEMS "

Development of scientific software involves tradeoffs between ease of use, generality, and performance. We describe the design of a general hyperbolic PDE solver that can be operated with the convenience of MATLAB yet achieves efficiency near that of hand-coded Fortran and scales to the largest supercomputers. This is achieved by using Python for most of the code while employing automatically wrapped Fortran kernels for computationally intensive routines, and using Python bindings to interface with a parallel computing library and other numerical packages. The software described here is PyClaw, a Python-based structured grid solver for general systems of hyperbolic PDEs [K. T. Mandli et al., PyClaw Software, Version 1.0, http://numerics.kaust.edu.sa/pyclaw/ (2011)]. PyClaw provides a powerful and intuitive interface to the algorithms of the existing Fortran codes Clawpack and SharpClaw, simplifying code development and use while providing massive parallelism and scalable solvers via the PETSc library. The...

High-Order Wave Propagation Algorithms for Hyperbolic Systems

We present a finite volume method that is applicable to hyperbolic PDEs including spatially varying and semilinear nonconservative systems. The spatial discretization, like that of the well-known Clawpack software, is based on solving Riemann problems and calculating fluctuations (not fluxes). The implementation employs weighted essentially nonoscillatory reconstruction in space and strong stability preserving Runge--Kutta integration in time. The method can be extended to arbitrarily high order of accuracy and allows a well-balanced implementation for capturing solutions of balance laws near steady state. This well-balancing is achieved through the

Optimized Explicit Runge--Kutta Schemes for the Spectral Difference Method Applied to Wave Propagation Problems

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

-wave Riemann solver and a novel wave-slope WENO reconstruction procedure. The wide applicability and advantageous properties of the method are demonstrated through numerical examples, including problems in nonconservative form, problems with spatially varying fluxes, and problems involving near-equilibrium solutions of balance laws.

Maximum-principle-satisfying space-time conservation element and solution element scheme applied to compressible multifluids

Explicit Runge-Kutta schemes with large stable step sizes are developed for integration of high order spectral difference spatial discretization on quadrilateral grids. The new schemes permit an effective time step that is substantially larger than the maximum admissible time step of standard explicit Runge-Kutta schemes available in literature. Furthermore, they have a small principal error norm and admit a low-storage implementation. The advantages of the new schemes are demonstrated through application to the Euler equations and the linearized Euler equations.

Entropy Stable Staggered Grid Discontinuous Spectral Collocation Methods of any Order for the Compressible Navier--Stokes Equations

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.

An Implicit Spectral Difference Navier-Stokes Solver For Unstructured Hexahedral Grids

A maximum-principle-satisfying space-time conservation element and solution element (CE/SE) scheme is constructed to solve a reduced five-equation model coupled with the stiffened equation of state for compressible multifluids. We first derive a sufficient condition for CE/SE schemes to satisfy maximum-principle when solving a general conservation law. And then we introduce a slope limiter to ensure the sufficient condition which is applicative for both central and upwind CE/SE schemes. Finally, we implement the upwind maximum-principle-satisfying CE/SE scheme to solve the volume-fraction-based five-equation model for compressible multifluids. Several numerical examples are carried out to carefully examine the accuracy, efficiency, conservativeness and maximum-principle-satisfying property of the proposed approach.

Large Eddy Simulation of a Muffler with the High-Order Spectral Difference Method

Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for the compressible Euler and Navier--Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel, SIAM J. Sci. Comput., 36 (2014), pp. B835--B867, M. Parsani, M. H. Carpenter, and E. J. Nielsen, J. Comput. Phys., 292 (2015), pp. 88--113], extends the applicable set of points from tensor product, Legendre--Gauss--Lobatto (LGL), to a combination of tensor product Legendre--Gauss (LG) and LGL points. The new semidiscrete operators discretely conserve mass, momentum, energy, and satisfy a mathematical entropy inequality for the compressible Navier--Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG ope...

Towards an Entropy Stable Spectral Element Framework for Computational Fluid Dynamics

A spectral difference (SD) solver using quadrilateral and hexahedral grids for the NavierStokes equations is presented. Three different approaches for the discretization of the diffusive terms with the SD method are tested and compared. These approaches are the local SD approach, the second approach of Bassi and Rebay and the interior penalty approach. Furthermore, two implicit methods to solve the nonlinear algebraic systems arising from SD discretizations have been implemented. The first is a Newton-Raphson method in combination with a generalized minimum residual algorithm to invert the associated linear algebraic systems. The second method is the (nonlinear) lower-upper symmetric Gauss-Seidel method. These two methods are used to solve the 2D laminar flow over a cylinder, and the 3D laminar flow through a pipe with a 90 ◦ bend, using the three different approaches for the diffusive terms. The obtained results are compared and the performance of the algebraic solvers in terms of CPU-time and memory is evaluated.

Efficiency of High Order Spectral Element Methods on Petascale Architectures

The combination of the high-order accurate spectral difference discretization on unstructured grids with subgrid-scale modelling is investigated for large eddy simulation of a muffler at Re = 4. 64 ⋅ 104 and M = 0. 05. The subgrid-scale stress tensor is modelled by the wall-adapting local eddy-viscosity model with a cut-off length which is a decreasing function of the order of accuracy of the scheme. Numerical results indicate that even when a fourth-order accurate scheme is used, the coupling with a subgrid-scale model improves the quality of the results.