Blended General Linear Methods based on Boundary Value Methods in the GBDF family

Among the methods for solving ODE-IVPs, the class of General Linear Methods (GLMs) is able to encompass most of them, ranging from Linear Multistep Formulae (LMF) to RK formulae. Moreover, it is possible to obtain methods able to overcome typical drawbacks of the previous classes of methods. For example, order barriers for stable LMF and the problem of order reduction for RK methods. Nevertheless, these goals are usually achieved at the price of a higher computational cost. Consequently, many efforts have been made in order to derive GLMs with particular features, to be exploited for their efficient implementation. In recent years, the derivation of GLMs from particular Boundary Value Methods (BVMs), namely the family of Generalized BDF (GBDF), has been proposed for the numerical solution of stiff ODE-IVPs. In particular, this approach has been recently developed, resulting in a new family of L-stable GLMs of arbitrarily high order, whose theory is here completed and fully worked-out. Moreover, for each one of such methods, it is possible to define a corresponding Blended GLM which is equivalent to it from the point of view of the stability and order properties. These blended methods, in turn, allow the definition of efficient nonlinear splittings for solving the generated discrete problems. A few numerical tests, confirming the excellent potential of such blended methods, are also reported.Comment: 22 pages, 8 figure

A fourth order symplectic and conjugate-symplectic extension of the midpoint and trapezoidal methods

The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite– Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and bound-ary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method

Quadrature formulas descending from BS Hermite spline quasi-interpolation

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Numerical Aspects of the Coefficient Computation for LMMs

The numerical solution of Boundary Value Problems usually requires the use of an adaptive mesh selection strategy. For this reason, when a Linear Multistep Method is considered, a dynamic computation of its coefficients is necessary. This leads to solve linear systems which can be expressed in different forms, depending on the polynomial basis used to impose the order conditions. In this paper, we compare the accuracy of the numerically computed coefficients for three different formulations. For all the considered cases Vandermonde systems on general abscissae are involved and they are always solved by the Bj \u308rk-Pereyra algorithm. An adaptation of the forward error analysis given in [8, 9] is proposed whose significance is confirmed by the numerical results

Some investigations into the numerical solution of initial value problems in ordinary differential equations

PhD ThesisIn this thesis several topics in the numerical solution of the initial value problem in first-order ordinary diff'erentlal equations are investigated, Consideration is given initially to stiff differential equations and their solution by stiffly-stable linear multistep methods which incorporate second derivative terms. Attempts are made to increase the size of the stability regions for these methods both by particular choices for the third characteristic polynomial and by the use of optimization techniques while investigations are carried out regarding the capabilities of a high order method. Subsequent work is concerned with the development of Runge-Kutta methods which include second-derivative terms and are implicit with respect to y rather than k. Methods of order three and four are proposed which are L-stable. The major part of the thesis is devoted to the establishment of recurrence relations for operators associated with linear multistep methods which are based on a non-polynomial representation of the theoretical solution. A complete set of recurrence relations is developed for both implicit and explicit multistep methods which are based on a representation involving a polynomial part and any number of arbitrary functions. The amount of work involved in obtaining mulc iste, :ne::l'lJds by this technique is considered and criteria are proposed to Jecide when this particular method of derivation should be em~loyed. The thesis is conclud~d by using Prony's method to develop one-step methods and multistep methods which are exponentially adaptive and as such can be useful in obtaining solutions to problems which are exponential in nature

An Accurate and Robust Numerical Scheme for Transport Equations

En esta tesis se presenta una nueva técnica de discretización para ecuaciones de transporte en problemas de convección-difusión para el rango completo de números de Péclet. La discretización emplea el flujo exacto de una ecuación de transporte unidimensional en estado estacionario para deducir una ecuación discreta de tres puntos en problemas unidimensionales y cinco puntos en problemas bidimensionales. Con "flujo exacto" se entiende que se puede obtener la solución exacta en función de integrales de algunos parámetros del fluido y flujo, incluso si estos parámetros son vari- ables en un volumen de control. Las cuadraturas de alto orden se utilizan para lograr resultados numéricos cercanos a la precisión de la máquina, incluso con mallas bastas.Como la discretización es esencialmente unidimensional, no está garantizada una solución con precisión de máquina para problemas multidimensionales, incluso en los casos en que las integrales a lo largo de cada coordenada cartesiana tienen una primitiva. En este sentido, la contribución principal de esta tesis consiste en una forma simple y elegante de obtener soluciones en problemas multidimensionales sin dejar de utilizar la formulación unidimensional. Además, si el problema es tal que la solución tiene precisión de máquina en el problema unidimensional a lo largo de las líneas coordenadas, también la tendrá para el dominio multidimensional.In this thesis, we present a novel discretization technique for transport equations in convection-diffusion problems across the whole range of Péclet numbers. The discretization employs the exact flux of a steady-state one-dimensional transport equation to derive a discrete equation with a three-point stencil in one-dimensional problems and a five-point stencil in two-dimensional ones. With "exact flux" it is meant that the exact solution can be obtained as a function of integrals of some fluid and flow parameters, even if these parameters are variable across a control volume. High-order quadratures are used to achieve numerical results close to machine- accuracy even with coarse grids. As the discretization is essentially one-dimensional, getting the machine- accurate solution of multidimensional problems is not guaranteed even in cases where the integrals along each Cartesian coordinate have a primitive. In this regard, the main contribution of this thesis consists in a simple and elegant way of getting solutions in multidimensional problems while still using the one-dimensional formulation. Moreover, if the problem is such that the solution is machine-accurate in the one-dimensional problem along coordinate lines, it will also be for the multidimensional domain.<br /

Numerical solutions of matrix differential models using higher-order matrix splines

The final publication is available at http://link.springer.com/article/10.1007%2Fs00009-011-0159-zThis paper deals with the construction of approximate solution of first-order matrix linear differential equations using higher-order matrix splines. An estimation of the approximation error, an algorithm for its implementation and some illustrative examples are included. © 2011 Springer Basel AG.Defez Candel, E.; Hervás Jorge, A.; Ibáñez González, JJ.; Tung, MM. (2012). Numerical solutions of matrix differential models using higher-order matrix splines. Mediterranean Journal of Mathematics. 9(4):865-882. doi:10.1007/s00009-011-0159-zS86588294Al-Said E.A., Noor M.A.: Cubic splines method for a system of third-order boundary value problems. Appl. Math. Comput. 142, 195–204 (2003)Ascher U., Mattheij R., Russell R.: Numerical solutions of boundary value problems for ordinary differential equations. Prentice Hall, New Jersey, USA (1988)Barnett S.: Matrices in Control Theory. Van Nostrand, Reinhold (1971)Blanes S., Casas F., Oteo J.A., Ros J.: Magnus and Fer expansion for matrix differential equations: the convergence problem. J. Phys. Appl. 31, 259–268 (1998)Boggs P.T.: The solution of nonlinear systems of equations by a-stable integration techniques. SIAM J. Numer. Anal. 8(4), 767–785 (1971)Defez E., Hervás A., Law A., Villanueva-Oller J., Villanueva R.: Matrixcubic splines for progressive transmission of images. J. Math. Imaging Vision 17(1), 41–53 (2002)Defez E., Soler L., Hervás A., Santamaría C.: Numerical solutions of matrix differential models using cubic matrix splines. Comput. Math. Appl. 50, 693–699 (2005)Defez E., Soler L., Hervás A., Tung M.M.: Numerical solutions of matrix differential models using cubic matrix splines II. Mathematical and Computer Modelling 46, 657–669 (2007)Mazzia F., Trigiante A.S., Trigiante A.S.: B-spline linear multistep methods and their conitinuous extensions. SIAM J. Numer. Anal. 44(5), 1954–1973 (2006)Faddeyev L.D.: The inverse problem in the quantum theory of scattering. J. Math. Physics 4(1), 72–104 (1963)Flett, T.M.: Differential Analysis. Cambridge University Press (1980)Golub G.H., Loan C.F.V.: Matrix Computations, second edn. The Johns Hopkins University Press, Baltimore, MD, USA (1989)Graham A.: Kronecker products and matrix calculus with applications. John Wiley & Sons, New York, USA (1981)Jódar L., Cortés J.C.: Rational matrix approximation with a priori error bounds for non-symmetric matrix riccati equations with analytic coefficients. IMA J. Numer. Anal. 18(4), 545–561 (1998)Jódar L., Cortés J.C., Morera J.L.: Construction and computation of variable coefficient sylvester differential problems. Computers Maths. Appl. 32(8), 41–50 (1996)Jódar, L., Ponsoda, E.: Continuous numerical solutions and error bounds for matrix differential equations. In: Int. Proc. First Int. Colloq. Num. Anal., pp. 73–88. VSP, Utrecht, The Netherlands (1993)Jódar L., Ponsoda E.: Non-autonomous riccati-type matrix differential equations: Existence interval, construction of continuous numerical solutions and error bounds. IMA J. Numer. Anal. 15(1), 61–74 (1995)Loscalzo F.R., Talbot T.D.: Spline function approximations for solutions of ordinary differential equations. SIAM J. Numer. Anal. 4(3), 433–445 (1967)Marzulli P.: Global error estimates for the standard parallel shooting method. J. Comput. Appl. Math. 34, 233–241 (1991)Micula G., Revnic A.: An implicit numerical spline method for systems for ode’s. Appl. Math. Comput. 111, 121–132 (2000)Reid, W.T.: Riccati Differential Equations. Academic Press (1972)Rektorys, K.: The method of discretization in time and partial differential equations. D. Reidel Pub. Co., Dordrecht (1982)Scott, M.: Invariant imbedding and its Applications to Ordinary Differential Equations. Addison-Wesley (1973