Thesis (Ph. D.)--Oregon State University, 1974. / Typescript (photocopy). Includes bibliographical references. Also available on the World Wide Web.

We derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent results and proposed numerical methods is given in chapter 2. Chapter 3 is devoted to the analysis of a family of implicit Chebyshev methods proposed by Panovsky k Richardson. We show that for each non-negative integer r, there are two methods of order 2r from this family which possess non-vanishing intervals of periodicity. The equivalence of these methods with one-step collocation methods is also established, and these methods are shown to be neither P-stable nor symplectic. In chapters 4 and 5, two families of multistep collocation methods are derived, and their order and stability properties are investigated. A detailed analysis of the two-step symmetric methods from each class is also given. The multistep Runge-Kutta-Nystrom methods of chapter 4 are found to be difficult to analyse, and the specific examples considered are found to perform poorly in the areas of both accuracy and stability. By contrast, the two-step symmetric hybrid methods of chapter 5 are shown to have excellent stability properties, in particular we show that all two-step 27V-point methods of this type possess non-vanishing intervals of periodicity, and we give conditions under which these methods are almost P-stable. P-stable and efficient methods from this family are obtained and demonstrated in numerical experiments. A simple, cheap and effective error estimator for these methods is also given.

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Propagation of radiation in a waveguides is theoretically described by Maxwell&amp / #8217 / s equations. The gradient of refractive index and an influence on the waveguide by a superstrate requires a numerical solution of the differential equation. Iterative methods such as the Runge-Kutta approaches are used to calculate the effective refractive index in the waveguide depending on the superstrate&amp / #8217 / s and the waveguide&amp / #8217 / s local refractive indices. In this study,the refractive indices, and the model fields of the TE00 modes are calculated. The calculated fields of the 00 TE modes give information about the propagation of the light in the waveguide. Also, the precision of the Runge-Kutta aproaches has been tested. The advantages and disadvantages of the Runge-Kutta aproaches are discussed.

Thèse de doctorat : Mécanique. Mécanique des Fluides : Strasbourg 1 : 2008. / Thèse soutenue sur un ensemble de travaux. Titre provenant de l'écran-titre. Notes bibliogr.

Numerical Weather Prediction (NWP) and climate models parametrize the effects of boundary-layer turbulence as a diffusive process, dependent on a diffusion coefficient, which appears as nonlinear terms in the governing equations. In the advection dominated zone of the boundary layer and in the free atmosphere, the air flow supports different wave motions, with the fastest being the sound waves. Time integrations of these terms, in both zones, need to be implicit otherwise they impractically restrict the stable time step sizes. At the same time, implicit schemes may lose accuracy compared to explicit schemes in the same level, which is due to dispersion error associated with these schemes. Furthermore, the implicit schemes need iterative approaches like the Newton-Raphson method. Therefore, the combination of implicit and explicit methods, called IMEX or semi-implicit, has extensively attracted attention. In the combined method, the linear part of the equation as well as the fast wave terms are treated by the implicit part and the rest is calculated by the explicit scheme. Meanwhile, minimizing the dissipation and dispersion errors can enhance the performance of time integration schemes, since the stability and accuracy will be restricted by these inevitable errors. Hence, the target of this thesis is to increase the stability range, while obtaining accurate solutions by using IMEX and semi-implicit time integration methods. Therefore, a comprehensive effort has been made toward minimizing the numerical errors to develop new Runge-Kutta schemes, in IMEX and semi-implicit forms, to temporally integrate the governing equations in the atmospheric field so that the stability is extended and accuracy is improved, compared to the previous schemes. At the first step, the A-stability and the Strong Stability Preserving (SSP) optimized properties were compared as two essential properties of the time integration schemes. It was shown that both properties attempt to minimize the dissipation and dispersion errors, but in two different aspects. The SSP optimized property focuses on minimizing the errors to increase the accuracy limits, while the A-stability property tries to extend the range of stability. It was shown that the combination of both properties is essential in the field of interest. Moreover, the A-stability property was found as an essential property to accelerate the steady state solutions. Afterward, the dissipation and dispersion errors, generated by three-stage second order IMEX Runge-Kutta scheme were minimized, while the proposed scheme, so called IMEX-SSP2(2,3,2) enjoys the A-stability and SSP properties. A practical governing equation set in the atmospheric field, so called compressible Boussinesq equations set, was calculated using the new IMEX scheme and the results were compared to one well-known IMEX scheme in the literature, i.e. ARK2(2,3,2), which is an abbreviation of Additive Runge-Kutta. Note that, the ARK2(2,3,2) was compared to various types of IMEX Runge-Kutta schemes and it was found as the more efficient scheme in the atmospheric fields (Weller et al., 2013). It was shown that the IMEX-SSP2(2,3,2) could improve the accuracy and extend the range of stable time step sizes as well. Through the van der Pol test case, it was shown that the ARK2(2,3,2) with L-stability property may decline to the first order in the calculation of stiff limit, while IMEX-SSP2(2,3,2), with A-stability property, is able to retain the assigned second order of accuracy. Therefore, it was concluded that the L-stability property, due to restrictive conditions associated with, may weaken the time integration’s performance, compared to the A-stability property. The ability of the IMEX-SSP2(2,3,2) was proved in solving different case, which is the inviscid Burger equation in spherical coordinate system by using a realistic initial condition dataset. In the next step, it was attempted to maximize the non-negativity property associated with the numerical stability function of three-stage third order Diagonally Implicit Runge-Kutta (DIRK) schemes. It was shown that the non-negativity has direct relation with non-oscillatory behaviors. Two new DIRK schemes with A- and L-stability properties, respectively, were developed and compared to the SSP(3,3), which obtains the SSP optimized property in the same class of DIRK schemes. The SSP optimized property was found to be more beneficial for the inviscid (advection dominated) flows, since in the von Neumann stability analysis, the SSP optimized property provides more nonnegative region for the imaginary component of the stability function. However, in most practical cases, i.e. the viscous (advection diffusion) flows, the nonnegative property is needed for both real and imaginary components of the stability function. Therefore, the SSP optimized property, individually, is not helpful, unless mixed with the A-stability property. Meanwhile, the A- and L-stability properties were compared as well. The intention is to find how these properties influence the DIRK schemes’ performances. The A-stability property was found as preserving the SSP property more than the L-stability property. Moreover, the proposed A-stable scheme tolerates larger Courant Friedrichs Lewy (CFL) number, while preserving the accuracy and non-oscillatory computations. This fact was proved in calculating different test cases, including compressible Euler and nonlinear viscous Burger equations. Finally, the time integration of the boundary layer flows was investigated as well. The nonlinearity associated with the diffusion coefficient makes the implicit scheme impractical, while the explicit scheme inefficiently limits the stable time step sizes. By using the DIRK scheme, a new semi-implicit approach was proposed, in which the diffusion coefficient at each internal stage is calculated by a weight-averaged combination of the solutions at current internal stage and previous time step, in which the time integration can benefit from both explicit and implicit advantages. As shown, the accuracy was improved, which is due to engaging the explicit solutions and the stability was extended due to taking advantages of implicit scheme. It was found that the nominated semi-implicit method results in less dissipation error, more accurate solutions and less CPU time usage, compared to the implicit schemes, and it enjoys larger range of stable time steps than other semi-implicit approaches in the literature.

This paper investigates the Runge-Kutta method of numerically integrating ordinary differential equations. An existence theorem is given insuring a solution to the differential equation, then the theorem is modified to yield an analytic solution. The derivation of the method itself is followed by an analysis of the inherent error.

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