We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods. Huynh, H. T. Glenn Research Center NASA/TM-2011-216340, E-17277

We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods.

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Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.

This thesis explores the ability of the discontinuous Galerkin (DG) method to numerically solve the Boltzmann equation. Constructing numerical methods for this equation is a challenge, due in part to the kinetic theory description of moving particles, which relies on space, time, and velocity variables. Two novel approaches are presented and compared. The first uses a spectral collocation basis in velocity space. The resulting system is solved in time using Diagonally Implicit Runge-Kutta methods, chosen in order to mitigate stiffness concerns. A Jacobian-Free Newton-Krylov method is presented, accelerated with a sweeping preconditioner. The method is tested on 1D and 2D problems in order to validate its convergence behavior and investigate its efficiency. The second method uses DG for moment equations, which can be derived as spectral methods in velocity space with spatial and temporal adaptivity. These methods were first proposed in 1949 by Grad, but their applicability has been limited. The equations are not guaranteed to be hyperbolic, leading to stability issues. The elegance and potential for cost-reduction of Grad's moment method have led to the development of different moment closures that preserve hyperbolicity and model accuracy. The approaches studied in this thesis, the globally hyperbolic moment methods, restore hyperbolicity by introducing a term that cannot be written in conservative form. The equations are typically solved with operator splitting and low-order methods. We examine the promise and challenges of applying a high-order DG method with explicit Runge-Kutta time-stepping to these equations on common 1D test cases. The thesis ends with a discussion on the prospects of both methods and suggestions for future work.

This book explains how to solve partial differential equations numerically using single and multidomain spectral methods. It shows how only a few fundamental algorithms form the building blocks of any spectral code, even for problems with complex geometries.