Entropy-stable residual distribution methods for system of euler equations
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Date
2016-09-01
Authors
Hossain Chizari
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Abstract
Residual-distribution (RD) methods have fundamental benefits over finite volume (FV)
or finite difference (FD) methods particularly in mimicking multi-dimensional physics,
achieving higher order accuracy with much smaller stencils and less sensitivity to
grid changes. The aim of this study is to develop a multi-dimensional entropy-stable
residual distribution method to solve the hyperbolic system of equations. First, an alternative
residual-distribution method is proposed to ensure conservation of primary
variables is obtained by default. This is followed by introducing a new signal distribution
and multi-dimensional entropy-conserved and entropy-stable RD method starting
with the two-dimensional Burgers’ equation. The development is extended to the
two-dimensional Euler equations. There will be rigorous mathematical analyses on
entropy-stability, multi-dimensional positivity, and truncation error study to determine
the formal order-of-accuracy for the entropy stable methods. In addition, it will also
be shown that conservation is automatic with the new RD method unlike with the current
RD methods where conservation requires a strict set of characteristic-averaging
within the elements and different systems of equations would require a different type
of averaging. The developments of limited entropy-stable RD methods would also
be included herein. Numerical experiments for the Burgers’ equation include an expansion
and a shock-tree problem followed by subsonic, transonic and supersonic gas
dynamics problem over various geometries for the Euler equations. Moreover, the
classical residual-distribution methods such as N, LDA, and PSI methods are studied
in this research to provide direct comparisons with the new entropy-stable RD methods.
Results of the new RD approach are comparable to the results of classic RD and
FV methods, yet are more robust for a variety of test cases.