86 research outputs found
Implicit High-Order Flux Reconstruction Solver for High-Speed Compressible Flows
The present paper addresses the development and implementation of the first
high-order Flux Reconstruction (FR) solver for high-speed flows within the
open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid
Dynamics) platform. The resulting solver is fully implicit and able to simulate
compressible flow problems governed by either the Euler or the Navier-Stokes
equations in two and three dimensions. Furthermore, it can run in parallel on
multiple CPU-cores and is designed to handle unstructured grids consisting of
both straight and curved edged quadrilateral or hexahedral elements. While most
of the implementation relies on state-of-the-art FR algorithms, an improved and
more case-independent shock capturing scheme has been developed in order to
tackle the first viscous hypersonic simulations using the FR method. Extensive
verification of the FR solver has been performed through the use of
reproducible benchmark test cases with flow speeds ranging from subsonic to
hypersonic, up to Mach 17.6. The obtained results have been favorably compared
to those available in literature. Furthermore, so-called super-accuracy is
retrieved for certain cases when solving the Euler equations. The strengths of
the FR solver in terms of computational accuracy per degree of freedom are also
illustrated. Finally, the influence of the characterizing parameters of the FR
method as well as the the influence of the novel shock capturing scheme on the
accuracy of the developed solver is discussed
Invariant preservation in machine learned PDE solvers via error correction
Machine learned partial differential equation (PDE) solvers trade the
reliability of standard numerical methods for potential gains in accuracy
and/or speed. The only way for a solver to guarantee that it outputs the exact
solution is to use a convergent method in the limit that the grid spacing
and timestep approach zero. Machine learned solvers,
which learn to update the solution at large and/or , can
never guarantee perfect accuracy. Some amount of error is inevitable, so the
question becomes: how do we constrain machine learned solvers to give us the
sorts of errors that we are willing to tolerate? In this paper, we design more
reliable machine learned PDE solvers by preserving discrete analogues of the
continuous invariants of the underlying PDE. Examples of such invariants
include conservation of mass, conservation of energy, the second law of
thermodynamics, and/or non-negative density. Our key insight is simple: to
preserve invariants, at each timestep apply an error-correcting algorithm to
the update rule. Though this strategy is different from how standard solvers
preserve invariants, it is necessary to retain the flexibility that allows
machine learned solvers to be accurate at large and/or .
This strategy can be applied to any autoregressive solver for any
time-dependent PDE in arbitrary geometries with arbitrary boundary conditions.
Although this strategy is very general, the specific error-correcting
algorithms need to be tailored to the invariants of the underlying equations as
well as to the solution representation and time-stepping scheme of the solver.
The error-correcting algorithms we introduce have two key properties. First, by
preserving the right invariants they guarantee numerical stability. Second, in
closed or periodic systems they do so without degrading the accuracy of an
already-accurate solver.Comment: 41 pages, 10 figure
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High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs
Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching
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Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws
The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling.
Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms
Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations
Discontinuous Galerkin (DG) methods have a long history in computational
physics and engineering to approximate solutions of partial differential
equations due to their high-order accuracy and geometric flexibility. However,
DG is not perfect and there remain some issues. Concerning robustness, DG has
undergone an extensive transformation over the past seven years into its modern
form that provides statements on solution boundedness for linear and nonlinear
problems.
This chapter takes a constructive approach to introduce a modern incarnation
of the DG spectral element method for the compressible Navier-Stokes equations
in a three-dimensional curvilinear context. The groundwork of the numerical
scheme comes from classic principles of spectral methods including polynomial
approximations and Gauss-type quadratures. We identify aliasing as one
underlying cause of the robustness issues for classical DG spectral methods.
Removing said aliasing errors requires a particular differentiation matrix and
careful discretization of the advective flux terms in the governing equations.Comment: 85 pages, 2 figures, book chapte
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