Two Interface-Type Numerical Methods for Computing Hyperbolic Systems with Geometrical Source Terms Having Concentrations

We propose two simple well-balanced methods for hyperbolic systems with geometrical source terms having concentrations. Physical problems under consideration include the shallow water equations with topography and the quasi-one-dimensional isothermal nozzle flows. These two methods use the numerical...

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Bibliographic Details
Published in:SIAM journal on scientific computing Vol. 26; no. 6; pp. 2079 - 2101
Main Authors: Jin, Shi, Wen, Xin
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
Subjects:
Accuracy
Applied mathematics
Computational methods in fluid dynamics
Exact sciences and technology
Fluid dynamics
Fundamental areas of phenomenology (including applications)
Mathematical analysis
Mathematics
Methods
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Physics
Sciences and techniques of general use
Topography
Values
Viscosity
Source terms
well-balanced schemes 35L65
Transonic flow
High resolution
Hyperbolic system
Isothermal flow
Supercritical flow
65M06
76B15
bottom topography
Unsteady state
Numerical method
Shallow water
Steady state
Numerical analysis
Scientific computation
Topography
shallow water equations
Well balanced scheme
Newton method
Computing method
Shallow-water equations
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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Summary:We propose two simple well-balanced methods for hyperbolic systems with geometrical source terms having concentrations. Physical problems under consideration include the shallow water equations with topography and the quasi-one-dimensional isothermal nozzle flows. These two methods use the numerical fluxes already obtained from the corresponding homogeneous systems in the source terms, and one needs only a black-box (approximate) Riemann solver for the homogeneous system. Compared with our previous method developed in [S. Jin and X. Wen, J. Comput. Math., 22 (2004), pp. 230--249], these methods avoid the Newton iterations in the evaluation of the source term. Numerical experiments demonstrate that both methods give good numerical approximations to the subcritical and supercritical flows. With a transonic fix, both methods also capture with a high resolution the transonic flows over the concentration. These methods are applicable to both unsteady and steady state computations.
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ISSN:1064-8275
1095-7197
DOI:10.1137/040605825