Are Gauss-Legendre methods useful in molecular dynamics?

Gauss-Legendre principal value integration

few problems lend themselves to closed-form solution, we often need to convert formal definitions into practical numerical methods. One such problem deals with the Principal Value integral, which many students encounter in a course on functions of a complex variable. However, the prospect of evaluating one numerically might seem rather daunting. To the best of my knowledge, the subject remains ...

Are Methods Really Useful?

Gauss Legendre-Gauss Jacobi quadrature rules over a tetrahedral region

This paper presents a Gaussian quadrature method for the evaluation of the triple integral ( , , ) T I f x y z d xd yd z = ∫∫∫ , where ) , , ( z y x f is an analytic function in , , x y z and T refers to the standard tetrahedral region:{( , , ) 0 , , 1, 1} x y z x y z x y z ≤ ≤ + + ≤ in three space( , , ). x y z Mathematical transformation from ( , , ) x y z space to ( , , ) u v w space maps th...

Filtering in Legendre spectral methods

We discuss the impact of modal filtering in Legendre spectral methods, both on accuracy and stability. For the former, we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity. Computational results confirm that less strict necessary conditions appear to be adequate. We proceed to discuss a instability mechanism in polynomial spectral method...

Two Point Gauss–legendre Quadrature Rule for Riemann–stieltjes Integrals

In order to approximate the Riemann–Stieltjes integral ∫ b a f (t) dg (t) by 2–point Gaussian quadrature rule, we introduce the quadrature rule ∫ 1 −1 f (t) dg (t) ≈ Af ( − √ 3 3 ) + Bf (√ 3 3 ) , for suitable choice of A and B. Error estimates for this approximation under various assumptions for the functions involved are provided as well.