On the dispersion-relation-preserving compact schemesor predicting electromagnetic wave, incompressible heat and fluid flow equations
Date Issued
2009
Date
2009
Author(s)
Chiu, Pao-Hsiung
Abstract
In this dissertation, schemes which accommodate a better dispersion relation for the convective terms shown in the transport equation, are proposed to enhance the convective stability by virtue of the increased dispersive accuracy.ll the schemes developed within this dissertation have been rigorously developed through the dispersion and dissipation analyses. To verify the proposed method, several problems that are amenable to the exact and benchmark solutions will be investigated. The results with good rates of convergence are demonstrated for all the investigated problems. The dispersion-relation-preserving schemes are then employed to nonlinear shallow water wave equation, electromagnetic wave equation, and incompressible fluid flow equation. For the nonlinear shallow water wave equation, we develop a computationally efficient and spatially accurate iterative scheme for solving the nonlinear shallow water wave equation, called Camassa-Holm (CH) equation. For computing an accurate dispersive solutions of the equation, we cast the equation in terms of an auxiliary variable to yield a linear convection-reaction equation that is coupled with a Helmholtz equation. We propose a sixth-order accurate advective schemehich accommodates the dispersion relation for the first-order spatial derivative term, while solving the coupled Helmholtz equation by a sixth-order accurate compact schemeormulated within a three-point stencil. In order to retain the Hamiltonian structure, a second-order-accurate and sixth-order-accurate symplectic time-stepping scheme is employed for the time integrator. Several test problems are provided, including a traveling wave solution, to justify the integrity of the proposed symplecticity- andispersion-relation-preserving discretization scheme. Comparisons between the proposed scheme, a complete integrable particle method, and the local discontinuous Galerkin method are also made in terms of accuracy,he elapsed computing time, and spatial and temporal rates of convergence. For the electromagnetic wave equation, a solver for electromagnetic wave equation, called Maxwell''s equations,s proposed in two- and three-dimensional non-staggered grids. To avoid even-odd spurious oscillations, the first-order spatial derivative terms will be approximated by the explicit compact scheme to save the computational time. To accommodate the Hamiltonian structure in the Maxwell''s equations, the time integrator employed in the current semi-discretization falls into the symplectic category. The integrity of the finite difference time domain method for solving the TM-mode Maxwell''s equations has been analytically verified through two- and three-dimensional test problems. For the incompressible fluid flow equation, we aim to develop a new formulation to effectively calculate the solutions of incompressible fluid flow equation, called Navier-Stokes equations, in non-staggered grids. The distinguished feature of the proposed method, which avoids directly solving the divergence-free equation, is to add a rigorously derived source term to the momentum equation to ensure satisfaction of the fluid incompressibility. For the sake of numerical accuracy, dispersion-relation-preserving upwind scheme developed within the two-dimensional context was employed to approximate the convection terms. The validity of the proposed mass-preserving Navier-Stokes method is justified by solving two benchmark problems at high Reynolds and Rayleigh numbers. Based on the simulated Navier-Stokes solutions,he proposed formulation is shown to outperform the onventional segregated method in terms of the reduction of CPU time. In order to solve the flow equations in irregular and time-varying domains, a immersed boundary (IB) method developed in Cartesian grids is applied together with the dispersion-relation-preserving dual-compact scheme. The artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body. We develop in this study a differential-based interpolation scheme which can be easily extended to three-dimensional simulation. The results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems. The accuracy and fidelity of the IB flow solver developed to predict flows with irregular boundaries are therefore demonstrated. Finally, we employ the present incompressible Naver-Stokes equations and run the code in message passing interface (MPI) parallel platform. With the domain decomposition methods combined with our divergence-free-condition compensated method, we can easily get higher performance for the present framework. The speed-up and efficiency are good for the simulation of lid-driven cavity problem.
Subjects
dispersion-relation-preserving
nonlinear swallow water wave
electromagnetic wave
incompressible flow
immersed boundary method
Type
thesis
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