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A detailed account of the stability and accuracy properties of the SBP-SAT technique for numerical time integration is presented. We show how the technique can be used to formulate both global and multi-stage methods with high order of accuracy for both stiff and non-stiff problems. Linear and nonlinear stability results, including A-stability, L-stability and B-stability are proven using the energy method for general initial value problems. Numerical experiments corroborate the theoretical properties.

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High frequency errors are always present in numerical simulations since no difference stencil is accurate in the vicinity of the π-mode. To remove the defective high wave number information from the solution, artificial dissipation operators or filter operators may be applied. Since stability is our main concern, we are interested in schemes on summation-by-parts (SBP) form with weak imposition of boundary conditions. Artificial dissipation operators preserving the accuracy and energy stability of SBP schemes are available. However, for filtering procedures it was recently shown that stability problems may occur, even for originally energy stable (in the absence of filtering) SBP based schemes. More precisely, it was shown that even the sharpest possible energy bound becomes very weak as the number of filtrations grow. This suggest that successful filtering include a delicate balance between the need to remove high frequency oscillations (filter often) and the need to avoid possible growth (filter seldom). We will discuss this problem and propose a remedy.