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Results for “euler equat”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

High-order limiting methods using maximum principle bounds derived from the Boltzmann equation I: Euler equations

The use of limiting methods for high-order numerical approximations of hyperbolic conservation laws generally requires defining an admissible region/bounds for the solution. In this work, we present a novel approach for computing solution bounds and limiting for the Euler equations through the kinetic representation provided by the Boltzmann equation, which allows for extending limiters designed for linear advection directly to the Euler equations. Given an arbitrary set of solution values to compute bounds over (e.g., numerical stencil) and a desired linear advection limiter, the proposed approach yields an analytic expression for the admissible region of particle distribution function values, which may be numerically integrated to yield a set of bounds for the density, momentum, and total energy. Further, these solution bounds are shown to preserve positivity of density/pressure/internal energy and, when paired with a limiting technique, can robustly resolve strong discontinuities while recovering high-order accuracy in smooth regions without any ad hoc corrections (e.g., relaxing the bounds). This approach is demonstrated in the context of an explicit unstructured high-order discontinuous Galerkin/flux reconstruction scheme for a variety of difficult problems in gas dynamics, including cases with extreme shocks and shock-vortex interactions. Furthermore, this work presents a foundation for limiting techniques for more complex macroscopic governing equations that can be derived from an underlying kinetic representation for which admissible solution bounds are not well-understood.

42 ENGINEERING↗
DOE OSTI2025

Symmetry Determining Equations of the Euler Equations

This purpose of this document is to provide a clear derivation of the symmetry determining equations (SDEs) of the Euler equations to supplement the lecture document. The Euler equations are a coupled system of three first-order quasilinear hyperbolic partial differential equations (PDEs). The Euler equations are a system of partial differential equations (PDEs) for fluid velocity v(t, r), fluid density ρ(t, r), and fluid pressure P(t, r), where t is time and r is the spatial coordinate.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Entropy viscosity method applied to Euler equations

The entropy viscosity method [4] has been successfully applied to hyperbolic systems of equations such as Burgers equation and Euler equations. The method consists in adding dissipative terms to the governing equations, where a viscosity coefficient modulates the amount of dissipation. The entropy viscosity method has been applied to the 1-D Euler equations with variable area using a continuous finite element discretization in the MOOSE framework and our results show that it has the ability to efficiently smooth out oscillations and accurately resolve shocks. Two equations of state are considered: Ideal Gas and Stiffened Gas Equations Of State. Results are provided for a second-order time implicit schemes (BDF2). Some typical Riemann problems are run with the entropy viscosity method to demonstrate some of its features. Then, a 1-D convergent-divergent nozzle is considered with open boundary conditions. The correct steady-state is reached for the liquid and gas phases with a time implicit scheme. The entropy viscosity method correctly behaves in every problem run. For each test problem, results are shown for both equations of state considered here. (authors)

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Euler equations and the Sod shock tube problem

The Euler equations are a subset of the magnetohydrodynamic (MHD) equations in the infinitely collisional, unmagnetized limit. MHD modeling is central to many areas of plasma physics, ranging from low-temperature glow discharges to inertial confinement fusion. An important aspect of the Euler equations is their ability to describe states with discontinuities, such as shock waves. A standard benchmark test for numerical implementation of the Euler equations is the Sod shock tube. In this test, the system is initialized at rest with a pressure and density discontinuity, which results in a shock wave traveling into the low-pressure region and a rarefaction wave traveling into the high-pressure region. Starting with the presentation of the Euler equations, a numerical algorithm is presented here to solve these equations in one dimension. This is followed by an overview of the Sod shock tube problem that includes the precise initial setup and the analytic solution. Finally, the analytic solution is compared with results from numerical simulations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2017

A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for xed values of the regularization parameter α > 0. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be suficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Scale Invariance of the Homentropic Inviscid Euler Equations with Application to the Noh Problem

We investigate the inviscid compressible flow (Euler) equations constrained by an ”isentropic” equation of state (EOS), whose functional form in pressure is an arbitrary function of density alone. Under the aforementioned condition, we interrogate using symmetry methods the scale-invariance of the homentropic inviscid Euler equations. We find that under general conditions, we can reduce the inviscid Euler equations into a system of two coupled ordinary differential equations. To exemplify the utility of these results, we formulate two example scale-invariant, self-similar solutions. The first example includes a shock-free expanding bubble scenario, featuring a modified Tait EOS. The second example features the classical Noh problem, coupled to an arbitrary isentropic EOS. In this case, in order to satisfy the conditions set forth in the classical Noh problem, we find that the solution for the flow is given by a transcendental algebraic equation in the shocked density.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A numerical extension of the spatially-filtered Euler equations for contact discontinuities

Solving the spatially-filtered Euler equations with a kinetic energy-preserving centered discretization is an efficient framework for simulating shock-dominated flows. Here, the present work describes a numerical treatment of the spatially-filtered Euler equations to minimize oscillations when simulating contact discontinuities. To counteract dispersive errors inherent to centered schemes, a WENO-like correction term is applied to the enthalpy transport in the energy equation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

A stochastic Galerkin method for the Euler equations with Roe variable transformation

The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion. In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Numerical solution of the Stratonovich- and Ito–Euler equations: Application to the stochastic piston problem

We consider a piston with a velocity perturbed by Brownian motion moving into a straight tube filled with a perfect gas at rest. The shock generated ahead of the piston can be located by solving the one-dimensional Euler equations driven by white noise using the Stratonovich or Ito formulations. We approximate the Brownian motion with its spectral truncation and subsequently apply stochastic collocation using either sparse grid or the quasi-Monte Carlo (QMC) method. In particular, we first transform the Euler equations with an unsteady stochastic boundary into stochastic Euler equations over a fixed domain with a time-dependent stochastic source term. We then solve the transformed equations by splitting them up into two parts, i.e., a ‘deterministic part’ and a ‘stochastic part’. Numerical results verify the Stratonovich–Euler and Ito–Euler models against stochastic perturbation results, and demonstrate the efficiency of sparse grid and QMC for small and large random piston motions, respectively. The variance of shock location of the piston grows cubically in the case of white noise in contrast to colored noise reported in [1], where the variance of shock location grows quadratically with time for short times and linearly for longer times.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

A sweeping positivity-preserving high-order finite difference WENO scheme for Euler equations

We develop a simple, high-order, conservative and robust positivity-preserving sweeping procedure for the density and the nonlinear pressure function in the compressible Euler equations. Using the scaling limiter in Zhang and Shu (J Comput Phys 229:3091–3120, 2010), we obtain a non-trivial extension of the scalar sweeping technique in Liu et al. (J Sci Comput 73:1028–1071, 2017) for the positivity of pressure. The sweeping procedure developed in this paper is a post-processing technique, which can be applied to any concave functions of the conserved variables in hyperbolic conservation law systems. Thus, it has applications beyond the Euler equations. This procedure preserves positivity and conservation of physical quantities without destroying the accuracy of the underlying scheme. The algorithm works for general schemes including finite difference, finite volume and discontinuous Galerkin methods; however, in this paper we focus on finite difference weighted essentially non-oscillatory (WENO) methods. As a result, we provide numerical tests of the fifth-order finite difference WENO scheme to demonstrate the accuracy and robustness of the technique.

Compressible Euler equations↗
DOE OSTI2014

Manufactured solutions for the three-dimensional Euler equations with relevance to Inertial Confinement Fusion

We present a set of manufactured solutions for the three-dimensional (3D) Euler equations. The purpose of these solutions is to allow for code verification against true 3D flows with physical relevance, as opposed to 3D simulations of lower-dimensional problems or manufactured solutions that lack physical relevance. Of particular interest are solutions with relevance to Inertial Confinement Fusion (ICF) capsules. While ICF capsules are designed for spherical symmetry, they are hypothesized to become highly 3D at late time due to phenomena such as Rayleigh–Taylor instability, drive asymmetry, and vortex decay. ICF capsules also involve highly nonlinear coupling between the fluid dynamics and other physics, such as radiation transport and thermonuclear fusion. The manufactured solutions we present are specifically designed to test the terms and couplings in the Euler equations that are relevant to these phenomena. Example numerical results generated with a 3D Finite Element hydrodynamics code are presented, including mesh convergence studies.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Robust second-order approximation of the compressible Euler equations with an arbitrary equation of state

Here, this paper is concerned with the approximation of the compressible Euler equations supplemented with an arbitrary or tabulated equation of state. The proposed approximation technique is robust, formally second-order accurate in space, invariant-domain preserving, and works for every equation of state, tabulated or analytic, provided the pressure is nonnegative. An entropy surrogate functional that grows across shocks is proposed. The numerical method is verified with novel analytical solutions and then validated with several computational benchmarks seen in the literature including problems with composite waves.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

On differentiable local bounds preserving stabilization for Euler equations

This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. Furthermore, the importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. Additionally, in the case of transient problem, we also observe a reduction in the computational cost.

42 ENGINEERING↗
DOE OSTI2024

A Performance-Portable MultiGPU Implementation of 3D Euler Equations using ProtoX and IRIS

Computational scientists often face challenges when developing and optimizing code for high-performance computing (HPC), especially when trying to leverage GPUs. Given the heterogeneity of the nodes that comprise many modern HPC facilities, considerable demand exists for performance portable solutions for the core computational kernels used in many scientific computing libraries. In this work, we demonstrate a fourth-order finite volume method–based implementation of the Euler equations, which are an integral part of computational fluid dynamics. Our performance-portable multiGPU implementation for Euler equations uses ProtoX to generate kernels and IRIS for portability. ProtoX is a domain-specific language that uses a structured-grid partial differential equation library called Proto as its front end and the SPIRAL code generation system as its back end to generate optimized kernels for different architectures. Optimized kernels generated by ProtoX are orchestrated through the IRIS intelligent runtime system to provide portability. Two levels of optimizations within the IRIS runtime— directed acyclic graph fusion and task fusion—are explored to efficiently utilize computing resources in a multiGPU environment. Performance improvement through these optimizations is showcased by comparing the base ProtoX-IRIS implementation on AMD GPUs (Frontier node) and on NVIDIA GPUs (NVIDIA DGX-1).

Mankad, Het↗
DOE OSTI2016

Second order upwind Lagrangian particle method for Euler equations

A new second order upwind Lagrangian particle method for solving Euler equations for compressible inviscid fluid or gas flows is proposed. Similar to smoothed particle hydrodynamics (SPH), the method represents fluid cells with Lagrangian particles and is suitable for the simulation of complex free surface / multiphase flows. The main contributions of our method, which is different from SPH in all other aspects, are (a) significant improvement of approximation of differential operators based on a polynomial fit via weighted least squares approximation and the convergence of prescribed order, (b) an upwind second-order particle-based algorithm with limiter, providing accuracy and long term stability, and (c) accurate resolution of states at free interfaces. In conclusion, numerical verification tests demonstrating the convergence order for fixed domain and free surface problems are presented.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Entropy–viscosity method for the single material Euler equations in Lagrangian frame

In this work, a new finite element method for solving the Euler equations in Lagrangian coordinates is proposed. The method is stabilized by adding artificial diffusion terms compatible with positivity of mass and internal energy, a minimum principle on the specific entropy, and all generalized entropy inequalities. Two options of first-order artificial diffusion are considered. One is in the spirit of the Eulerian based method (Guermond et al., 2011 [23, 22]; Zingan et al., 2013) and the other is similar to existing viscosity stabilizations in Lagrangian frame, e.g., Dobrev et al. (2012). The method is verified to be high-order for smooth solutions even with active viscosity terms. This is achieved by using high-order finite element spaces and an entropy-based viscosity stabilization that degenerates the first-order viscous terms. This stabilization automatically distinguishes smooth and singular regions. The formal accuracy and convergence properties of the proposed methods are tested on a series of benchmark problems. This is the first result extending the entropy–viscosity methodology to the Lagrangian hydrodynamics.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Continuous properties of the solution map for the Euler equations

In this paper, we study the dependence on initial data of solutions to the incompressible Euler equations in Besov spaces. We show that for s > n/p + 1, p ∈ (1, ∞), and r ∈ [1, ∞], the solution map u{sub 0}↦u is Hölder continuous in Besov space B{sub p,r}{sup s} equipped with weaker topology. When the space variable x is taken to be periodic, we obtain a family of explicit periodic solutions. Furthermore, we prove that for any s∈R and 1 ⩽ r ⩽ ∞, the solution map is not globally uniformly continuous in B{sub 2,r}{sup s}(T{sup n}), which extends some results of Himonas and Misiołek [Commun. Math. Phys. 296(1), 285–301 (2010)].

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

High-Fidelity, Low-Dissipation/Symmetry-Preserving Numerical Scheme for Solving the Euler Equations with Unstructured, Metric-Based Mesh Adaptation

This work presents an overview of a high-fidelity compressible Euler solver that utilizes the continuous Galerkin (CG) method with added artificial numerical diffusion for stabilization to solve a variety of unsteady and steady benchmark inviscid flow problems. This work shows that discretizing the Euler equations with this CG approach and first order basis functions produces a cost-effective stencil as well as simple well-posed boundary conditions. We show through convergence testing with manufactured solutions that the reduced stencil of CG, combined with the low amount of artificial diffusion required when using the stabilization method outlined in this work, leads to stable and highly accurate results for a variety of unsteady and steady applications. When combined with the adaptive mesh refinement approach used for many of the cases in this work, our results show that the flow solver achieves even more accurate results. A variety of inviscid flow cases are presented in this work, including transient 2D cases with complex shock structures and several steady 3D airfoils sections with a constant span.

Doetsch, Kevin [ORNL] (ORCID:0000000267051705)↗