Rates of convergence (or divergence) are obtained in the application of Gauss, Lobatto, and Radau integration rules to functions with an algebraic or logarithmic singularity inside, or at an endpoint of, the interval of integration. A typical result is the following: For a generalized Jacobi weight function on [-1,1], the error in applying an «-point rule to f(x) = \x -y\~* isO(n~2 + 2i), if y ...

in this paper, a chebyshev finite difference method has been proposed in order to solvenonlinear two-point boundary value problems for second order nonlinear differentialequations. a problem arising from chemical reactor theory is then considered. the approachconsists of reducing the problem to a set of algebraic equations. this method can be regardedas a non-uniform finite difference scheme. t...

Several types of simultaneous approximation term (SAT) for diffusion problems discretized with diagonal-norm multidimensional summation-by-parts (SBP) operators are analyzed based on a common framework. Conditions under which the SBP-SAT discretizations consistent, conservative, adjoint and energy stable presented. For SATs leading to primal consistent discretizations, error in output functiona...

The RK5GL3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on a combination of a fifth-order Runge-Kutta method and 3-point Gauss-Legendre quadrature. In this paper we describe the propagation of local errors in this method, and show that the global order of RK5GL3 is expected to be six, one better than the underlying RungeKutta m...

Abstract. Advection of trace species, or tracers, also called tracer transport, in models the atmosphere and other physical domains is an important potentially computationally expensive part a model's dynamical core. Semi-Lagrangian (SL) advection methods are efficient because they permit time step much larger than advective stability limit for explicit Eulerian without requiring solution globa...

A number of conservative PDEs, like various wave equations, allow for a multi-symplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. We show that Gauss-Legendre collocation in space and time leads to multi-symplectic integrators, i.e., to numerical methods that preserve a symplectic conservation law similar to the conservation of symplect...

In this paper, double integrals over an arbitrary quadrilateral are evaluated exploiting finite element method. The physical region is transformed into a standard quadrilateral finite element using the basis functions in local space. Then the standard quadrilateral is subdivided into two triangles, and each triangle is further discretized into 4 n right isosceles triangles, with area 1 2n2, and...

Filon-Simpson quadrature rules are derived for integrals of the type ∫ b a dx f(x) sin(xy)/(xy) and ∫ b a dx f(x) 4 sin(xy/2)/(xy) which are needed in applications of the worldline variational approach to Quantum Field Theory. These new integration rules reduce to the standard Simpson rule for y = 0 and are exact for y → ∞ when a = 0 and f(0) 6= 0. The subleading term in the asymptotic expansio...