arXiv subjects

Matteo Parsani

Publications and source records attributed to Matteo Parsani.

30 records · Page 2Linked to original sources

arXiv2019

Optimized geometrical metrics satisfying free-stream preservation

Computational fluid dynamics and aerodynamics, which complement more expensive empirical approaches, are critical for developing aerospace vehicles. During the past three decades, computational aerodynamics capability has improved remarkably, following advances in computer hardware and algorithm development. However, for complex applications, the demands on computational fluid dynamics continue to increase in a quest to gain a few percent improvements in accuracy. Herein, we numerically demonstrate that optimizing the metric terms which arise from smoothly mapping each cell to a reference element, lead to a solution whose accuracy is practically never worse and often noticeably better than the one obtained using the widely adopted Thomas and Lombard metric terms computation (Geometric conservation law and its application to flow computations on moving grids, AIAA Journal, 1979). Low and high-order accurate entropy stable schemes on distorted, high-order tensor product elements are used to simulate three-dimensional inviscid and viscous compressible test cases for which an analytical solution is known.

math.NA↗

arXiv2019

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 (David I. Ketcheson, \emph{Relaxation Runge-Kutta Methods: Conservation and Stability for Inner-Product Norms}, SIAM Journal on Numerical Analysis, 2019) is extended to general convex quantities. Conservation, dissipation, or other solution properties with respect to any convex functional are enforced by the addition of a {\em relaxation parameter} that multiplies the Runge-Kutta update at each step. Moreover, other desirable stability (such as strong stability preservation) and efficiency (such as low storage requirements) properties are preserved. The technique can be applied to both explicit and implicit Runge-Kutta methods and requires only a small modification to existing implementations. The computational cost at each step is the solution of one additional scalar algebraic equation for which a good initial guess is available. The effectiveness of this approach 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.

math.NA↗

arXiv2019

More efficient time integration for Fourier pseudo-spectral DNS of incompressible turbulence

Time integration of Fourier pseudo-spectral DNS is usually performed using the classical fourth-order accurate Runge--Kutta method, or other methods of second or third order, with a fixed step size. We investigate the use of higher-order Runge-Kutta pairs and automatic step size control based on local error estimation. We find that the fifth-order accurate Runge--Kutta pair of Bogacki \& Shampine gives much greater accuracy at a significantly reduced computational cost. Specifically, we demonstrate speedups of 2x-10x for the same accuracy. Numerical tests (including the Taylor-Green vortex, Rayleigh-Taylor instability, and homogeneous isotropic turbulence) confirm the reliability and efficiency of the method. We also show that adaptive time stepping provides a significant computational advantage for some problems (like the development of a Rayleigh-Taylor instability) without compromising accuracy.

math.NA↗

arXiv2019

Entropy Stable h/p-Nonconforming Discretization with the Summation-by-Parts Property for the Compressible Euler and Navier-Stokes Equations

In this paper, the entropy conservative/stable algorithms presented by Del Rey Fernandez and coauthors [18,16,17] for the compressible Euler and Navier-Stokes equations on nonconforming p-refined/coarsened curvilinear grids is extended to h/p refinement/coarsening. The main difficulty in developing nonconforming algorithms is the construction of appropriate coupling procedures across nonconforming interfaces. Here, a computationally simple and efficient approach based upon using decoupled interpolation operators is utilized. The resulting scheme is entropy conservative/stable and element-wise conservative. Numerical simulations of the isentropic vortex and viscous shock propagation confirm the entropy conservation/stability and accuracy properties of the method (achieving ~ p + 1 convergence) which are comparable to those of the original conforming scheme [4,35]. Simulations of the Taylor-Green vortex at Re = 1,600 and turbulent flow past a sphere at Re = 2,000 show the robustness and stability properties of the overall spatial discretization for unstructured grids. Finally, to demonstrate the entropy conservation property of a fully-discrete explicit entropy stable algorithm with h/p refinement/coarsening, we present the time evolution of the entropy function obtained by simulating the propagation of the isentropic vortex using a relaxation Runge-Kutta scheme.

math.NA↗

arXiv2019

Entropy Stable p-Nonconforming Discretizations with the Summation-by-Parts Property for the Compressible Navier-Stokes Equations

The entropy conservative, curvilinear, nonconforming, p-refinement algorithm for hyperbolic conservation laws of Del Rey Fernandez et al. (2019), is extended from the compressible Euler equations to the compressible Navier-Stokes equations. A simple and flexible coupling procedure with planar interpolation operators between adjoining nonconforming elements is used. Curvilinear volume metric terms are numerically approximated via a minimization procedure and satisfy the discrete geometric conservation law conditions. Distinct curvilinear surface metrics are used on the adjoining interfaces to construct the interface coupling terms, thereby localizing the discrete geometric conservation law constraints to each individual element. The resulting scheme is entropy conservative/stable, element-wise conservative, and freestream preserving. Viscous interface dissipation operators are developed that retain the entropy stability of the base scheme. The accuracy and stability properties of the resulting numerical scheme are shown to be comparable to those of the original conforming scheme (achieving ~p+1 convergence) in the context of the viscous shock problem, the Taylor-Green vortex problem at a Reynolds number of Re=1,600, and a subsonic turbulent flow past a sphere at Re = 2,000.

math.NA↗

arXiv2019

Conservative and entropy stable solid wall boundary conditions for the compressible Navier-Stokes equations: Adiabatic wall and heat entropy transfer

We present a novel technique for the imposition of non-linear entropy conservative and entropy stable solid wall boundary conditions for the compressible Navier-Stokes equations in the presence of an adiabatic wall, or a wall with a prescribed heat entropy flow. The procedure relies on the formalism and mimetic properties of diagonal-norm, summation-by-parts, and simultaneous-approximation-term operators, and is a generalization of previous works on discontinuous interface coupling [1] and solid wall boundary conditions [2]. Using the method of lines, a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions are coupled with an entropy-conservative or entropy-stable discrete interior operator. The resulting estimate mimics the global entropy estimate obtained at the continuous level. The boundary data at the wall are weakly imposed using a penalty flux approach and a simultaneous-approximation-term technique for both the conservative variables and the gradient of the entropy variables. Discontinuous spectral collocation operators (mass lumped nodal discontinuous Galerkin operators), on high-order unstructured grids, are used for the purpose of demonstrating the robustness and efficacy of the new procedure for weakly enforcing boundary conditions. Numerical simulations confirm the non-linear stability of the proposed technique, with applications to three-dimensional subsonic and supersonic flows. The procedure described is compatible with any diagonal-norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction schemes.

math.NA↗

arXiv2017

An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property

This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre-Gauss-Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between nonconforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general non-linear hyperbolic systems on h/p non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples.

math.NA↗

arXiv2014

Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier--Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.

physics.flu-dyn↗

arXiv2014

Propagation of internal errors in explicit Runge--Kutta methods and internal stability of SSP and extrapolation methods

In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.

math.NA↗

arXiv2013

Large eddy simulation of a muffler with the high-order spectral difference method

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 10^4 and low Mach number. 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 although the high-order solver without subgrid-scale modelling is already able to capture well the features of the flow, the coupling with the wall-adapting local eddy-viscosity model improves the quality of the solution.

physics.flu-dyn↗

arXiv2012

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 non-oscillatory 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 $f$-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.

math.NA↗

arXiv2012

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 \cite{pyclaw}. 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 package is further augmented by use of PyWENO for generation of efficient high-order weighted essentially non-oscillatory reconstruction code. The simplicity, capability, and performance of this approach are demonstrated through application to example problems in shallow water flow, compressible flow and elasticity.

math.NA↗