48,469 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
Higher order finite difference schemes for the magnetic induction equations
We describe high order accurate and stable finite difference schemes for the
initial-boundary value problem associated with the magnetic induction
equations. These equations model the evolution of a magnetic field due to a
given velocity field. The finite difference schemes are based on Summation by
Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation
Term (SAT) technique for imposing boundary conditions. We present various
numerical experiments that demonstrate both the stability as well as high order
of accuracy of the schemes.Comment: 20 page
Convergence of summation-by-parts finite difference methods for the wave equation
In this paper, we consider finite difference approximations of the second
order wave equation. We use finite difference operators satisfying the
summation-by-parts property to discretize the equation in space. Boundary
conditions and grid interface conditions are imposed by the
simultaneous-approximation-term technique. Typically, the truncation error is
larger at the grid points near a boundary or grid interface than that in the
interior. Normal mode analysis can be used to analyze how the large truncation
error affects the convergence rate of the underlying stable numerical scheme.
If the semi-discretized equation satisfies a determinant condition, two orders
are gained from the large truncation error. However, many interesting second
order equations do not satisfy the determinant condition. We then carefully
analyze the solution of the boundary system to derive a sharp estimate for the
error in the solution and acquire the gain in convergence rate. The result
shows that stability does not automatically yield a gain of two orders in
convergence rate. The accuracy analysis is verified by numerical experiments.Comment: In version 2, we have added a new section on the convergence analysis
of the Neumann problem, and have improved formulations in many place
On discretely entropy conservative and entropy stable discontinuous Galerkin methods
High order methods based on diagonal-norm summation by parts operators can be
shown to satisfy a discrete conservation or dissipation of entropy for
nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as
nodal discontinuous Galerkin methods with diagonal mass matrices. In this work,
we describe how use flux differencing, quadrature-based projections, and
SBP-like operators to construct discretely entropy conservative schemes for DG
methods under more arbitrary choices of volume and surface quadrature rules.
The resulting methods are semi-discretely entropy conservative or entropy
stable with respect to the volume quadrature rule used. Numerical experiments
confirm the stability and high order accuracy of the proposed methods for the
compressible Euler equations in one and two dimensions
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