Impulse formulation of the Euler equations: general properties and numerical methods

1999 ◽  
Vol 391 ◽  
pp. 189-209 ◽  
Author(s):  
GIOVANNI RUSSO ◽  
PETER SMEREKA
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Gauge Transformation ◽  
Euler Equations ◽  
Numerical Scheme ◽  
Numerical Results ◽  
Gauge Freedom ◽  
General Properties ◽  
Three Space ◽  
Incompressible Euler

The gauge freedom of the incompressible Euler equations is explored. We present various forms of the Euler equations written in terms of the impulse density. It is shown that these various forms are related by a gauge transformation. We devise a numerical method to solve the impulse form of the Euler equations in a variety of gauges. The numerical scheme is implemented both in two and three space dimensions. Numerical results are presented showing that the impulse density tends to concentrate on sheets.

scholarly journals Solving Dual Fuzzy Nonlinear Equations via Shamanskii Method

2018 ◽  
Vol 7 (3.28) ◽  
pp. 89 ◽  
Author(s):  
Ibrahim Mohammed Sulaiman ◽  
Mustafa Mamat ◽  
Nurnadiah Zamri ◽  
Puspa Liza Ghazali
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Nonlinear Equations ◽  
Computational Cost ◽  
Numerical Results ◽  
Parametric Form ◽  
Solution Point ◽  
New Ideas

New ideas on numerical methods for solving fuzzy nonlinear equations have spread quickly across the globe. However, most of the methods available are based on Newton’s approach whose performance is impaired by either discontinuity or singularity of the Jacobian at the solution point. Also, the study of dual fuzzy nonlinear equations is yet to be explored by many researchers. Thus, in this paper, a numerical method to investigate the solution of dual fuzzy nonlinear equations is proposed. This method reduces the computational cost of Jacobian evaluation at every iteration. The fuzzy coefficients are presented in its parametric form. Numerical results obtained have shown that the proposed method is efficient. 

High Order Numerical Methods for Fractional Terminal Value Problems

2014 ◽  
Vol 14 (1) ◽  
pp. 55-70 ◽  
Author(s):  
Neville J. Ford ◽  
Maria L. Morgado ◽  
Magda Rebelo
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Numerical Scheme ◽  
Initial Value ◽  
Shooting Algorithm ◽  
Effective Approximation

Abstract. In this paper we present a shooting algorithm to solve fractional terminal (or boundary) value problems. We provide a convergence analysis of the numerical method, derived based upon properties of the equation being solved and without the need to impose smoothness conditions on the solution. The work is a sequel to our recent investigation where we constructed a nonpolynomial collocation method for the approximation of the solution to fractional initial value problems. Here we show that the method can be adapted for the effective approximation of the solution of terminal value problems. Moreover, we compare the efficiency of this numerical scheme against other existing methods.

Symplectic Irregular Interpolation Algorithms for Optimal Control Problems

2015 ◽  
Vol 12 (06) ◽  
pp. 1550040 ◽  
Author(s):  
Mingwu Li ◽  
Haijun Peng ◽  
Zhigang Wu
Keyword(s):  
Optimal Control ◽  
Numerical Methods ◽  
Numerical Method ◽  
Computational Efficiency ◽  
Optimal Control Problems ◽  
Numerical Results ◽  
Control Problems ◽  
Collocation Points ◽  
Comparison Results ◽  
Made In

Symplectic numerical methods for optimal control problems with irregular interpolation schemes are developed and the comparisons between irregular interpolation schemes and equidistant scheme are made in this paper. The irregular interpolation points, which are the collocation points usually adopted by pseudospectral (PS) methods, such as Legendre–Gauss, Legendre–Gauss–Radau, Legendre–Gauss–Lobatto and Chebyshev–Gauss–Lobatto points, are taken into consideration in this study. The symplectic numerical method with irregular points is proposed firstly. Then, several examples with different complexities highlight the differences in performance between different kinds of interpolation schemes. The numerical results show that the convergence of the present symplectic numerical methods can be obtained by increasing the number of sub-intervals or the number of interpolation points. Moreover, the comparison results show that the convergence of the symplectic numerical methods are generally independent on the type of interpolation points and the computational efficiency is not sensitive to the choice of interpolation points in general. Thus, the symplectic numerical methods with different interpolation schemes have obvious difference with the PS methods.

The Zero-Mach Limit of Compressible Euler Equations

2013 ◽  
Vol 275-277 ◽  
pp. 518-521
Author(s):  
Shu Wang
Keyword(s):  
Mach Number ◽  
Euler Equations ◽  
Initial Velocity ◽  
Compressible Flows ◽  
Incompressible Flows ◽  
Compressible Euler Equations ◽  
Time Interval ◽  
Compressible Euler ◽  
Three Space ◽  
Incompressible Euler

The aim of this paper is to prove that compressible Euler equations in two and three space dimensions converge to incompressible Euler equations in the limit as the Mach number tends to zero. No smallness restrictions are imposed on the initial velocity, or the time interval. We assume instead that the incompressible flows exists and is reasonably smooth on a given time interval, and prove that compressible flows converge uniformly on that time interval.

scholarly journals Circulation and Energy Theorem Preserving Stochastic Fluids

2019 ◽  
Vol 150 (6) ◽  
pp. 2776-2814 ◽  
Author(s):  
Theodore D. Drivas ◽  
Darryl D. Holm
Keyword(s):  
Variational Principle ◽  
Energy Conservation ◽  
Euler Equations ◽  
Navier Stokes ◽  
Fluid Models ◽  
Smooth Solutions ◽  
Stochastic Fluid Models ◽  
Natural Extensions ◽  
Stochastic Fluid ◽  
Incompressible Euler

AbstractSmooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

scholarly journals Numerical methods for solving the multi-term time-fractional wave-diffusion equation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Fawang Liu ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Pinghui Zhuang ◽  
Qingxia Liu
Keyword(s):  
Numerical Methods ◽  
Theoretical Analysis ◽  
Diffusion Equation ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Time Space ◽  
Methods And Techniques ◽  
Wave Diffusion

AbstractIn this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Observations Regarding Numerical Results Obtained by the Floquet-Method

2013 ◽  
Author(s):  
Till J. Kniffka ◽  
Horst Ecker
Keyword(s):  
Numerical Methods ◽  
Monodromy Matrix ◽  
Mathieu Equation ◽  
Numerical Results ◽  
The Other ◽  
Benchmark Problem ◽  
Stability Studies ◽  
Floquet Method ◽  
System Equations ◽  
Time Periodic

Stability studies of parametrically excited systems are frequently carried out by numerical methods. Especially for LTP-systems, several such methods are known and in practical use. This study investigates and compares two methods that are both based on Floquet’s theorem. As an introductary benchmark problem a 1-dof system is employed, which is basically a mechanical representation of the damped Mathieu-equation. The second problem to be studied in this contribution is a time-periodic 2-dof vibrational system. The system equations are transformed into a modal representation to facilitate the application and interpretation of the results obtained by different methods. Both numerical methods are similar in the sense that a monodromy matrix for the LTP-system is calculated numerically. However, one method uses the period of the parametric excitation as the interval for establishing that matrix. The other method is based on the period of the solution, which is not known exactly. Numerical results are computed by both methods and compared in order to work out how they can be applied efficiently.

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