12,676 research outputs found
An Explicit Fourth-Order Hybrid-Variable Method for Euler Equations with A Residual-Consistent Viscosity
In this paper we present a formally fourth-order accurate hybrid-variable
method for the Euler equations in the context of method of lines. The
hybrid-variable (HV) method seeks numerical approximations to both
cell-averages and nodal solutions and evolves them in time simultaneously; and
it is proved in previous work that these methods are inherent superconvergent.
Taking advantage of the superconvergence, the method is built on a third-order
discrete differential operator, which approximates the first spatial derivative
at each grid point, only using the information in the two neighboring cells.
Stability and accuracy analyses are conducted in the one-dimensional case for
the linear advection equation; whereas extension to nonlinear systems including
the Euler equations is achieved using characteristic decomposition and the
incorporation of a residual-consistent viscosity to capture strong
discontinuities. Extensive numerical tests are presented to assess the
numerical performance of the method for both 1D and 2D problems.Comment: 26 page
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
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