117 research outputs found
An explicit high-order single-stage single-step positivity-preserving finite difference WENO method for the compressible Euler equations
In this work we construct a high-order, single-stage, single-step
positivity-preserving method for the compressible Euler equations. Space is
discretized with the finite difference weighted essentially non-oscillatory
(WENO) method. Time is discretized through a Lax-Wendroff procedure that is
constructed from the Picard integral formulation (PIF) of the partial
differential equation. The method can be viewed as a modified flux approach,
where a linear combination of a low- and high-order flux defines the numerical
flux used for a single-step update. The coefficients of the linear combination
are constructed by solving a simple optimization problem at each time step. The
high-order flux itself is constructed through the use of Taylor series and the
Cauchy-Kowalewski procedure that incorporates higher-order terms. Numerical
results in one- and two-dimensions are presented
An a posteriori, efficient, high-spectral resolution hybrid finite-difference method for compressible flows
Versión aceptada de https://doi.org/10.1016/j.cma.2018.02.013[Abstract:] A high-order hybrid method consisting of a high-accurate explicit finite-difference scheme and a Weighted Essentially Non-Oscillatory (WENO) scheme is proposed in this article. Following this premise, two variants are outlined: a hybrid made up of a Finite Difference scheme and a compact WENO scheme (CRWENO 5), and a hybrid made up of a Finite Difference scheme and a non-compact WENO scheme (WENO 5). The main difference with respect to similar schemes is its a posteriori nature, based on the Multidimensional Optimal Order Detection (MOOD) method. To deal with complex geometries, a multi-block approach using Moving Least Squares (MLS) procedure for communication between meshes is used. The hybrid schemes are validated with several 1D and 2D test cases to illustrate their accuracy and shock-capturing properties.This work has been partially supported by the Ministerio de Economía y Competitividad (grant #DPI2015-68431-R) of the Spanish Government and by the Consellería de Educación e Ordenación Universitaria of the Xunta de Galicia (grant #GRC2014/039), cofinanced with FEDER funds and the Universidade da Coruña.Xunta de Galicia; GRC2014/03
Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers
In this paper we use the genuinely multidimensional HLL Riemann solvers
recently developed by Balsara et al. to construct a new class of
computationally efficient high order Lagrangian ADER-WENO one-step ALE finite
volume schemes on unstructured triangular meshes. A nonlinear WENO
reconstruction operator allows the algorithm to achieve high order of accuracy
in space, while high order of accuracy in time is obtained by the use of an
ADER time-stepping technique based on a local space-time Galerkin predictor.
The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the
grid, considering the entire Voronoi neighborhood of each node and allows for
larger time steps than conventional one-dimensional Riemann solvers. The
results produced by the multidimensional Riemann solver are then used twice in
our one-step ALE algorithm: first, as a node solver that assigns a unique
velocity vector to each vertex, in order to preserve the continuity of the
computational mesh; second, as a building block for genuinely multidimensional
numerical flux evaluation that allows the scheme to run with larger time steps
compared to conventional finite volume schemes that use classical
one-dimensional Riemann solvers in normal direction. A rezoning step may be
necessary in order to overcome element overlapping or crossing-over. We apply
the method presented in this article to two systems of hyperbolic conservation
laws, namely the Euler equations of compressible gas dynamics and the equations
of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to
fourth order of accuracy in space and time have been carried out. Several
numerical test problems have been solved to validate the new approach
A Physical-Constraint-Preserving Finite Volume WENO Method for Special Relativistic Hydrodynamics on Unstructured Meshes
This paper presents a highly robust third-order accurate finite volume
weighted essentially non-oscillatory (WENO) method for special relativistic
hydrodynamics on unstructured triangular meshes. We rigorously prove that the
proposed method is physical-constraint-preserving (PCP), namely, always
preserves the positivity of the pressure and the rest-mass density as well as
the subluminal constraint on the fluid velocity. The method is built on a
highly efficient compact WENO reconstruction on unstructured meshes, a simple
PCP limiter, the provably PCP property of the Harten--Lax--van Leer flux, and
third-order strong-stability-preserving time discretization. Due to the
relativistic effects, the primitive variables (namely, the rest-mass density,
velocity, and pressure) are highly nonlinear implicit functions in terms of the
conservative variables, making the design and analysis of our method
nontrivial. To address the difficulties arising from the strong nonlinearity,
we adopt a novel quasilinear technique for the theoretical proof of the PCP
property. Three provable convergence-guaranteed iterative algorithms are also
introduced for the robust recovery of primitive quantities from admissible
conservative variables. We also propose a slight modification to an existing
WENO reconstruction to ensure the scaling invariance of the nonlinear weights
and thus to accommodate the homogeneity of the evolution operator, leading to
the advantages of the modified WENO reconstruction in resolving multi-scale
wave structures. Extensive numerical examples are presented to demonstrate the
robustness, expected accuracy, and high resolution of the proposed method.Comment: 56 pages, 18 figure
Recent developments in accuracy and stability improvement of nonlinear filter methods for DNS and LES of compressible flows
Recent progress in the improvement of numerical stability and accuracy of the Yee and Sjögreen [49] high order nonlinear filter schemes is described. The Yee & Sjögreen adaptive nonlinear filter method consists of a high order non-dissipative spatial base scheme and a nonlinear filter step. The nonlinear filter step consists of a flow sensor and the dissipative portion of a high resolution nonlinear high order shock-capturing method to guide the application of the shock-capturing dissipation where needed. The nonlinear filter idea was first initiated by Yee et al. [54] using an artificial compression method (ACM) of Harten [12] as the flow sensor. The nonlinear filter step was developed to replace high order linear filters so that the same scheme can be used for long time integration of direct numerical simulations (DNS) and large eddy simulations (LES) for both shock-free turbulence and turbulence-shock waves inter- actions. The improvement includes four major new developments: (a) Smart flow sensors were developed to replace the global ACM flow sensor [21,22,50]. The smart flow sensor provides the locations and the estimated strength of the necessary numerical dissipation needed at these locations and leaves the rest of the flow field free of shock-capturing dissipation. (b) Skew-symmetric splittings were developed for compressible gas dynamics and magnetohydrodynamics (MHD) equations [35,36] to improve numerical stability for long time integration. (c) High order entropy stable numerical fluxes were developed as the spatial base schemes for both the compressible gas dynamics and MHD [37,38]. (d) Several dispersion relation-preserving (DRP) central spatial schemes were included as spatial base schemes in the frame- work of our nonlinear filter method approach [40]. With these new scheme constructions the nonlinear filter schemes are applicable to a wider class of accurate and stable DNS and LES applications, including forced turbulence simulations where the time evolution of flows might start with low speed shock-free turbulence and develop into supersonic speeds with shocks. Representative test cases for both smooth flows and problems containing discontinuities for compressible flows are included
Dissipation-based WENO stabilization of high-order finite element methods for scalar conservation laws
We present a new perspective on the use of weighted essentially
nonoscillatory (WENO) reconstructions in high-order methods for scalar
hyperbolic conservation laws. The main focus of this work is on nonlinear
stabilization of continuous Galerkin (CG) approximations. The proposed
methodology also provides an interesting alternative to WENO-based limiters for
discontinuous Galerkin (DG) methods. Unlike Runge--Kutta DG schemes that
overwrite finite element solutions with WENO reconstructions, our approach uses
a reconstruction-based smoothness sensor to blend the numerical viscosity
operators of high- and low-order stabilization terms. The so-defined WENO
approximation introduces low-order nonlinear diffusion in the vicinity of
shocks, while preserving the high-order accuracy of a linearly stable baseline
discretization in regions where the exact solution is sufficiently smooth. The
underlying reconstruction procedure performs Hermite interpolation on stencils
consisting of a mesh cell and its neighbors. The amount of numerical
dissipation depends on the relative differences between partial derivatives of
reconstructed candidate polynomials and those of the underlying finite element
approximation. All derivatives are taken into account by the employed
smoothness sensor. To assess the accuracy of our CG-WENO scheme, we derive
error estimates and perform numerical experiments. In particular, we prove that
the consistency error of the nonlinear stabilization is of the order ,
where is the polynomial degree. This estimate is optimal for general
meshes. For uniform meshes and smooth exact solutions, the experimentally
observed rate of convergence is as high as
An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs
We extend the entropy stable high order nodal discontinuous Galerkin spectral
element approximation for the non-linear two dimensional shallow water
equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J.
Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin
method for the two dimensional shallow water equations on unstructured
curvilinear meshes with discontinuous bathymetry. Journal of Computational
Physics, 340:200-242, 2017] with a shock capturing technique and a positivity
preservation capability to handle dry areas. The scheme preserves the entropy
inequality, is well-balanced and works on unstructured, possibly curved,
quadrilateral meshes. For the shock capturing, we introduce an artificial
viscosity to the equations and prove that the numerical scheme remains entropy
stable. We add a positivity preserving limiter to guarantee non-negative water
heights as long as the mean water height is non-negative. We prove that
non-negative mean water heights are guaranteed under a certain additional time
step restriction for the entropy stable numerical interface flux. We implement
the method on GPU architectures using the abstract language OCCA, a unified
approach to multi-threading languages. We show that the entropy stable scheme
is well suited to GPUs as the necessary extra calculations do not negatively
impact the runtime up to reasonably high polynomial degrees (around ). We
provide numerical examples that challenge the shock capturing and positivity
properties of our scheme to verify our theoretical findings
Skew-symmetric splitting of high-order central schemes with nonlinear filters for computational aeroacoustics turbulence with shocks
A class of high-order nonlinear filter schemes by Yee et al. (J Comput Phys 150:199–238, 1999), Sjögreen and Yee (J Comput Phys 225:910–934, 2007), and Kotov et al. (Commun Comput Phys 19:273–300, 2016; J Comput Phys 307:189–202, 2016) is examined for long-time integrations of computational aeroacoustics (CAA) turbulence applications. This class of schemes was designed for an improved nonlinear stability and accuracy for long-time integration of compressible direct numerical simulation and large eddy simulation computations for both shock-free turbulence and turbulence with shocks. They are based on the skew-symmetric splitting version of the high-order central base scheme in conjunction with adaptive low-dissipation control via a nonlinear filter step to help with stability and accuracy capturing at shock-free regions as well as in the vicinity of discontinuities. The central dispersion-relation-preserving schemes as well as classical central schemes of arbitrary orders fit into the framework of skew-symmetric splitting of the inviscid flux derivatives. Numerical experiments on CAA turbulence test cases are validated
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