A Class of Hybrid DG/FV Methods for Conservation Laws III: Two-Dimensional Euler Equations

2012 ◽  
Vol 12 (1) ◽  
pp. 284-314 ◽  
Author(s):  
Laiping Zhang ◽  
Wei Liu ◽  
Lixin He ◽  
Xiaogang Deng
Keyword(s):  
Euler Equations ◽  
Lower Order ◽  
Wave Interaction ◽  
Weak Shock ◽  
Higher Order ◽  
Two Dimensional ◽  
Dynamic Reconstruction ◽  
Third Order ◽  
Dg Method ◽  
Hybrid Grids

AbstractA concept of “static reconstruction” and “dynamic reconstruction” was introduced for higher-order (third-order or more) numerical methods in our previous work. Based on this concept, a class of hybrid DG/FV methods had been developed for one-dimensional conservation law using a “hybrid reconstruction” approach, and extended to two-dimensional scalar equations on triangular and Cartesian/triangular hybrid grids. In the hybrid DG/FV schemes, the lower-order derivatives of the piece-wise polynomial are computed locally in a cell by the traditional DG method (called as “dynamic reconstruction”), while the higher-order derivatives are re-constructed by the “static reconstruction” of the FV method, using the known lower-order derivatives in the cell itself and in its adjacent neighboring cells. In this paper, the hybrid DG/FV schemes are extended to two-dimensional Euler equations on triangular and Cartesian/triangular hybrid grids. Some typical test cases are presented to demonstrate the performance of the hybrid DG/FV methods, including the standard vortex evolution problem with exact solution, isentropic vortex/weak shock wave interaction, subsonic flows past a circular cylinder and a three-element airfoil (30P30N), transonic flow past a NACA0012 airfoil. The accuracy study shows that the hybrid DG/FV method achieves the desired third-order accuracy, and the applications demonstrate that they can capture the flow structure accurately, and can reduce the CPU time and memory requirement greatly than the traditional DG method with the same order of accuracy.

Scaling of detonation velocity in cylinder and slab geometries for ideal, insensitive and non-ideal explosives

2015 ◽  
Vol 773 ◽  
pp. 224-266 ◽  
Author(s):  
Scott I. Jackson ◽  
Mark Short
Keyword(s):  
Phase Velocity ◽  
Detonation Velocity ◽  
Lower Order ◽  
Higher Order ◽  
Slab Thickness ◽  
Two Dimensional ◽  
Order Model ◽  
Scaling Behaviour ◽  
Geometric Scaling ◽  
Shock Curvature

Experiments were conducted to characterize the detonation phase-velocity dependence on charge thickness for two-dimensional detonation in condensed-phase explosive slabs of PBX 9501, PBX 9502 and ANFO. In combination with previous diameter-effect measurements from a cylindrical rate-stick geometry, these data permit examination of the relative scaling of detonation phase velocity between axisymmetric and two-dimensional detonation. We find that the ratio of cylinder radius ($R$) to slab thickness ($T$) at each detonation phase velocity ($D_{0}$) is such that $R(D_{0})/T(D_{0})<1$. The variation in the $R(D_{0})/T(D_{0})$ scaling is investigated with two detonation shock dynamics (DSD) models: a lower-order model relates the normal detonation velocity to local shock curvature, while a higher-order model includes the effect of front acceleration and transverse flow. The experimentally observed $R(D_{0})/T(D_{0})$ (${<}1$) scaling behaviour for PBX 9501 and PBX 9502 is captured by the lower-order DSD theory, revealing that the variation in the scale factor is due to a difference in the slab and axisymmetric components of the curvature along the shock in the cylindrical geometry. The higher-order DSD theory is required to capture the observed $R(D_{0})/T(D_{0})$ (${<}1$) scaling behaviour for ANFO. An asymptotic analysis of the lower-order DSD formulation describes the geometric scaling of the detonation phase velocity between the cylinder and slab geometries as the detonation phase velocity approaches the Chapman–Jouguet value.

Higher-order schemes for 3D first-arrival traveltimes and amplitudes

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. T47-T56 ◽  
Author(s):  
Songting Luo ◽  
Jianliang Qian ◽  
Hongkai Zhao
Keyword(s):  
Point Source ◽  
Helmholtz Equation ◽  
Higher Order ◽  
Two Dimensional ◽  
Order Convergence ◽  
Third Order ◽  
First Order ◽  
First Arrival ◽  
Function Smooth ◽  
Good Agreement

In the geometrical-optics approximation for the Helmholtz equation with a point source, traveltimes and amplitudes have upwind singularities at the point source. Hence, both first-order and higher-order finite-difference solvers exhibit formally at most first-order convergence and relatively large errors. Such singularities can be factored out by factorizing traveltimes and amplitudes, where one factor is specified to capture the corresponding source singularity and the other factor is an unknown function smooth near the source. The resulting underlying unknown functions satisfy factored eikonal and transport equations, respectively. A third-order Lax-Friedrichs scheme is designed to compute the underlying functions. Thus, highly accurate first-arrival traveltimes and reliable amplitudes can be computed. Furthermore, asymptotic wavefields are constructed using computed traveltimes and amplitudes in the WKBJ form. Two-dimensional and 3D examples demonstrate the performance of the proposed algorithms, and the constructed WKBJ Green’s functions are in good agreement with direct solutions of the Helmholtz equation before caustics occur.

On higher-order wave theory for submerged two-dimensional bodies

1969 ◽  
Vol 38 (2) ◽  
pp. 415-432 ◽  
Author(s):  
Nils Salvesen
Keyword(s):  
Free Surface ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Drag ◽  
Higher Order ◽  
Second Order ◽  
Two Dimensional ◽  
Third Order ◽  
Free Surface Waves ◽  
Body Shapes

The importance of non-linear free-surface effects on potential flow past two-dimensional submerged bodies is investigated by the use of higher-order perturbation theory. A consistent second-order solution for general body shapes is derived. A comparison between experimental data and theory is presented for the free-surface waves and for the wave resistance of a foil-shaped body. The agreement is good in general for the second-order theory, while the linear theory is shown to be inadequate for predicting the wave drag at the relatively small submergence treated here. It is also shown, by including the third-order freesurface effects, how the solution to the general wave theory breaks down at low speeds.

scholarly journals An Analytic Evolutionary Algorithm To Compute Higher-Order Solutions For Model Equation

2021 ◽  
Author(s):  
Bang-Qing Li ◽  
Yu-Lan Ma
Keyword(s):  
Schrödinger Equation ◽  
Evolutionary Algorithm ◽  
Lower Order ◽  
Schrodinger Equation ◽  
Model Equation ◽  
Evolutionary Computing ◽  
Higher Order ◽  
Third Order ◽  
Generalized Schrödinger Equation ◽  
Basic Features

Abstract In this work, by introducing Darboux operator in evolutionary computing frame, we propose a novel analytic evolutionary algorithm to obtain exact higher-order iteration solutions for model equation. We construct the first-, second- and third-order solutions of a generalized Schrödinger equation by applying this algorithm. The higher-order solutions not only retain the basic features of the lower-order cases, but also become more abundant than the lower-order cases.

A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids

2012 ◽  
Vol 12 (5) ◽  
pp. 1495-1519 ◽  
Author(s):  
Hong Luo ◽  
Luqing Luo ◽  
Robert Nourgaliev
Keyword(s):  
Galerkin Method ◽  
Euler Equations ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Least Squares Method ◽  
Polynomial Solution ◽  
Third Order ◽  
Von Neumann ◽  
Flow Problems ◽  
Dg Method

AbstractA reconstruction-based discontinuous Galerkin (RDG(P1P2)) method, a variant of P1P2 method, is presented for the solution of the compressible Euler equations on arbitrary grids. In this method, an in-cell reconstruction, designed to enhance the accuracy of the discontinuous Galerkin method, is used to obtain a quadratic polynomial solution (P2) from the underlying linear polynomial (P1) discontinuous Galerkin solution using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG(P1P2) method is third-order accurate, and outperforms the third-order DG method (DG(P2)) in terms of both computing costs and storage requirements.

Rogue-Wave Interaction of a Nonlinear Schrödinger Model for the Alpha Helical Protein

2016 ◽  
Vol 71 (1) ◽  
pp. 27-32 ◽  
Author(s):  
Hui-Xian Jia ◽  
Yu-Jun Liu ◽  
Ya-Ning Wang
Keyword(s):  
Rogue Wave ◽  
Wave Interaction ◽  
Rogue Waves ◽  
Higher Order ◽  
Order Effects ◽  
Third Order ◽  
Nonlinear Schrödinger ◽  
Existence Time ◽  
Helical Protein ◽  
Higher Order Effects

AbstractIn this article, we investigate a fourth-order nonlinear Schrödinger equation, which governs the Davydov solitons in the alpha helical protein with higher-order effects. By virtue of the generalised Darboux transformation, higher-order rogue-wave solutions are derived. Propagation and interaction of the rogue waves are analysed: (i) Coefficients affect the existence time of the first-order rogue waves; (ii) coefficients affect the interaction time of the second- and third-order rogue waves; (iii) direction of the rogue-wave propagation remain unchanged after interaction.

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