Jukka Tuomela

On singular points of quasilinear differential and differential-algebraic equations

We show how certain singularities of quasilinear differential and differential-algberaic equations can be resolved by taking the solutions to be integral manifolds of certain distributions rather than curves with specific parametrization.

On the construction of arbitrary order schemes for the manydimensional wave equation

We show that it is possible to construct arbitrary order stable schemes for the homogeneous and heterogeneous wave equation in any dimension. The construction is elementary and uses formal power series techniques. We shall also calculate exact stability limits in various cases, and apparently this limit depends only on the dimension of the space.

A note on high order schemes for the one dimensional wave equation

We construct in two ways stable schemes of arbitrarily high order for one dimensional wave equation. In both cases the stability limit is independent of the order. The other construction yields also automatically stable schemes for the heterogeneous case. In this case the estimated stability limit decreases slightly with order.

On the resolution of singularities of ordinary differential systems

We show how some differential geometric ideas help to resolve some singularities of ordinary differential systems. Hence a singular problem is replaced by a regular one, which facilitates further analysis of the system. The methods employed are constructive and the regularized systems can also be used for numerical computations.

Algebraic derivation of discrete absorbing boundary conditions for the wave equation

Summary. We introduce a new algebraic framework to derive discrete absorbing boundary conditions for the wave equation in the one-dimensional case. The idea is to factor directly the discrete wave operator and then use one of the factors as boundary condition. We also analyse the stability of the schemes obtained this way and perform numerical simulations to estimate their practical value.