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Port, Valve, Intake, and Exhaust System Design
Published in John B. Heywood, Eran Sher, The Two-Stroke Cycle Engine, 2017
The method of characteristics is a well-established mathematical technique for solving hyperbolic partial differential equations. With this technique, the partial differential equations are transferred into ordinary differential equations that apply along so-called characteristic lines. Pressure waves are the physical phenomenon of practical interest in the unsteady intake flow, and these propagate relative to the flowing gas at the local sound speed. In this particular application, the one-dimensional unsteady flow equations are rearranged so that they contain only the local fluid velocity U and local sound speed c.
The Finite Difference Method
Published in Victor N. Kaliakin, Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods, 2018
Hyperbolic partial differential equations have two real and distinct characteristics. In regions defined by these characteristics, information travels at finite speeds. Hyperbolic partial differential equations arise in several problems of engineering interest, such as vibration of strings and rods, transmission of sound in air, and the flow of an object in a stationary fluid or of the fluid past stationary objects at speeds greater than the speed of sound in that fluid (i.e., supersonic flow).
Basic Relations
Published in M. Necati Özişik, Helcio R.B. Orlande, Marcelo José Colaço, Renato Machado Cotta, Finite Difference Methods in Heat Transfer, 2017
M. Necati Özişik, Helcio R.B. Orlande, Marcelo José Colaço, Renato Machado Cotta
Problems governed by hyperbolic partial differential equations are encountered in a number of applications in heat and fluid flow. For example, transient heat conduction associated with laser pulses of extremely short duration, extremely high rates of change of temperature or heat fluxes, or extremely low temperatures approaching absolute zero may be governed by the hyperbolic heat conduction equation (1.6) instead of by the customarily used parabolic heat conduction equation (1.4).
Determination of a time-dependent potential in the higher-order pseudo-hyperbolic problem
Published in Inverse Problems in Science and Engineering, 2021
Pseudo-hyperbolic partial differential equations appear in the modelling of various phenomena such as medicine, biology, mineral exploration, seismology, quality control in industry, etc. The inverse initial boundary value problems for the pseudo-hyperbolic equations with the different types of boundary conditions and with the time or space-dependent coefficients have been drawing a growing attention due to its wide range of applications. For instance, Lorenzi and Paparoni [1] considered the identification problems for recovering the kernels in a pseudo-hyperbolic integro-differential operator equation. The inverse problem for recovering coefficients of hyperbolic equations, with time-dependent/independent unknown coefficients, have been investigated by the authors of [2–4]. For example, Bellassoued and Yamamoto [5] determined the coefficient of inverse problem for the second-order hyperbolic equation with variable coefficients. In [6,7], the time-dependent coefficients are reconstructed for the hyperbolic equation. The study was continued by Eskin [3] to identify the time-dependent coefficient for the second-order general hyperbolic equations. In [2,8], the authors investigated the inverse problems to determine the unknown coefficients in the second-order hyperbolic equations. In Hussein et al. [9], the authors investigated the identification of the spacewise potential and/or damping functions from Cauchy data boundary measurements.
A Divergence-Free High-Order Spectral Difference Method with Constrained Transport for Ideal Compressible Magnetohydrodynamics
Published in International Journal of Computational Fluid Dynamics, 2021
In this paper, an unstaggered constrained transport (CT) framework proposed by Christlieb, Rossmanith, and Tang (2014) is used to satisfy the divergence-free condition in the discrete sense. Here a brief review of this framework is given. First, the divergence-free magnetic field is written as the curl of the magnetic vector potential Plugging Equation (36) into Equation (33), we obtain Using the Weyl gauge proposed in Helzel, Rossmanith, and Taetz (2011), the governing equation of the magnetic vector potential becomes For two-dimensional problems, it reduces to It is a hyperbolic partial differential equation. Therefore the spectral difference method can be a suitable method to solve it. The at element interfaces is calculated as its upwind value.
High-Order Lax-Friedrichs WENO Fast Sweeping Methods for the SN Neutron Transport Equation
Published in Nuclear Science and Engineering, 2019
Dean Wang, Tseelmaa Byambaakhuu
The development of robust numerical solutions for hyperbolic conservation laws has been a major research effort in the last few decades. For steady-state problems of hyperbolic conservation laws, a natural approach to computing these stationary solutions is to use an explicit-time-stepping or pseudo-time-stepping technique to evolve the system to steady state. However, the computational efficiency of these schemes is restricted by the Courant-Friedrichs-Lewy condition and needs substantial time to reach the steady-state solution. Fast sweeping methods are efficient iterative numerical schemes originally designed for solving stationary Hamilton-Jacobi equations.1 In Ref. 2, Chen et al. generalize the fast sweeping methods to hyperbolic conservation laws with source terms. The numerical solution algorithm was obtained through finite difference discretization, with the Lax-Friedrichs (LF) numerical fluxes evaluated in Weighted Essentially Non-Oscillatory (WENO) fashion, coupled with Gauss-Seidel iterations. WENO finite difference/volume schemes are a popular class of high-order numerical methods for solving hyperbolic partial differential equations (PDEs). They have the advantage of attaining uniform high-order accuracy in smooth regions of the solution while maintaining sharp and essentially monotone transitions of discontinuities.3,4