Abstract.A new weak boundary procedure for hyperbolic problems is presented. We consider high order finite difference operators of summation-by-parts form with weak boundary conditions and generalize that technique. The new boundary procedure is applied near boundaries in an extended domain where data is known. We show how to raise the order of accuracy of the scheme, how to modify the spectrum of the resulting operator and how to construct non-reflecting properties at the boundaries. The new boundary procedure is cheap, easy to implement and suitable for all numerical methods, not only finite difference methods, that employ weak boundary conditions. Numerical results that corroborate the analysis are presented.

A one dimensional steady-state advection-diffusion problem using summationby-parts operators has been investigated. For approximating the second derivative, a wide stencil has been used, which has spurious, oscillating, modes for all mesh-sizes. We show that the size of the spurious modes are equal to the size of the truncation error for a stable approximation. The theoretical results are verified with numerical experiments.

Temporal error bounds for the wave equation expressed on second order form are investigated. We show that, with appropriate choices of boundary conditions, the time and space derivatives of the error are bounded even for long times. No long time bound on the error itself is obtained, although numerical experiments indicate that a bound exists.

A lateral boundary treatment using summation-by-parts operators and simultaneous approximation terms is introduced. The method is similar to Davies relaxation technique used in the weather prediction community and have similar areas of application, but is also provably stable. In this paper, it is shown how this technique can be applied to the shallow water equations, and that it reduces the errors in the computational domain.

In this paper we provide a new approach for constructing non-reflecting boundary conditions. The boundary conditions are based on summation-by-parts operators and derived without Laplace transformation in time. We prove that the new non-reflecting boundary conditions yield a well-posed problem and that the corresponding numerical approximation is unconditionally stable. The analysis is demonstrated on a hyperbolic system in two space dimensions, and the theoretical results are confirmed by numerical experiments