159 research outputs found
Review of Summation-by-parts schemes for initial-boundary-value problems
High-order finite difference methods are efficient, easy to program, scales
well in multiple dimensions and can be modified locally for various reasons
(such as shock treatment for example). The main drawback have been the
complicated and sometimes even mysterious stability treatment at boundaries and
interfaces required for a stable scheme. The research on summation-by-parts
operators and weak boundary conditions during the last 20 years have removed
this drawback and now reached a mature state. It is now possible to construct
stable and high order accurate multi-block finite difference schemes in a
systematic building-block-like manner. In this paper we will review this
development, point out the main contributions and speculate about the next
lines of research in this area
Spectral methods for the wave equation in second-order form
Current spectral simulations of Einstein's equations require writing the
equations in first-order form, potentially introducing instabilities and
inefficiencies. We present a new penalty method for pseudo-spectral evolutions
of second order in space wave equations. The penalties are constructed as
functions of Legendre polynomials and are added to the equations of motion
everywhere, not only on the boundaries. Using energy methods, we prove
semi-discrete stability of the new method for the scalar wave equation in flat
space and show how it can be applied to the scalar wave on a curved background.
Numerical results demonstrating stability and convergence for multi-domain
second-order scalar wave evolutions are also presented. This work provides a
foundation for treating Einstein's equations directly in second-order form by
spectral methods.Comment: 16 pages, 5 figure
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