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We focus on the solution of multiparameter spectral problems, and in particular on some strategies to compute coarse approximations of selected eigenparameters depending on the number of oscillations of the associated eigenfunctions. Since the computation of the eigenparameters is crucial in codes for multiparameter problems based on finite differences, we herein present two strategies. The first one is an iterative algorithm computing solutions as limit of a set of decoupled problems (much easier to solve). The second one solves problems depending on a parameter \(\sigma \in [0,1]\), that give back the original problem only when \(\sigma =1\). We compare the strategies by using well known test problems with two and three parameters.

In this work, we discuss the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform, important for numerous applications. Due to the very special behavior of the Hankel transform eigenfunctions, their direct numerical calculation often causes an essential loss of accuracy. Here, we present several simple, efficient and robust numerical techniques to compute Hankel transform eigenfunctions via the associated singular self-adjoint Sturm-Liouville operator. The properties of the proposed approaches are compared and illustrated by means of numerical experiments.

We discuss an initial value problem for an implicit second order ordinary differential equation which arises in models of flow in saturated porous media such as concrete. Depending on the initial condition, the solution features a sharp interface with derivatives which become numerically unbounded. By using an integrator based on finite difference methods and equipped with adaptive step size selection, it is possible to compute the solution on highly irregular meshes. In this way it is possible to verify and predict asymptotical theory near the interface with remarkable accuracy.

In this note we give implementation details on the computation of the monitor function which is used inside a code based on high order upwind methods. The considered strategy is based on deferred correction and allows to compute an approximation of the error for a method of order p with essential no additional computational cost.