Exponential fitted Runge–Kutta methods of collocation type: fixed or variable knot points?

Space-Time Discretization of Hyperbolic Equations with Variable Collocation Points

In the field of scalar hyperbolic equations, we take into consideration the idea of building numerical schemes using two grids: the first one to represent the solution and the second one to collocate the equation. This approach is based on the experience already gained in the approximation of boundaryvalue elliptic-type problems (see for instance [1]), as well as in the field of functional ot i...

Counting Fixed Points, Two-Cycles, and Collisions of the Discrete Exponential Function using p-adic Methods

Brizolis asked for which primes p greater than 3 does there exist a pair (g, h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Zhang (1995) and Cobeli and Zaharescu (1999) answered with a “yes” for sufficiently large primes and gave estimates for the number of such pairs when g and h are primiti...

Iterative Methods for Equilibrium Problems, Variational Inequalities and Fixed Points

Exponential Rosenbrock-Type Methods

We introduce a new class of exponential integrators for the numerical integration of large-scale systems of stiff differential equations. These so-called Rosenbrock-type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the Jacobian. In contrast to standard integrators, the methods are fully explicit and do not require the numerical sol...

Effective order strong stability preserving RungeKutta methods

We apply the concept of effective order to strong stability preserving (SSP) explicit Runge–Kutta methods. Relative to classical Runge–Kutta methods, effective order methods are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. The relaxed order conditions allow for greater freedom in the design of ef...