Weak solutions for partial differential equations with source terms: Application to the shallow water equations

Journal of Computational Physics

Volumn 229, Issue 11, 2010, Pages 4327-4368

  Murillo, J ^a   Garcia Navarro P ^a  

^a UNIVERSITY OF ZARAGOZA   (Spain)

Author keywords

Energy dissipating solutions; Roe methods; Shallow water systems; Stability region; Weak solutions; Well balanced approach; Wet dry front

Indexed keywords

EQUATIONS OF MOTION; PARTIAL DIFFERENTIAL EQUATIONS;

ENERGY DISSIPATING; ENERGY DISSIPATING SOLUTION; ROE'S METHODS; SHALLOW-WATER SYSTEMS; SOURCE TERMS; STABILITY REGIONS; WEAK SOLUTION; WELL BALANCED; WELL-BALANCED APPROACH; WET/DRY FRONTS;

SYSTEM STABILITY;

EID: 77950041338     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.02.016     Document Type: Article

References (21)

1. Akanbi A.A., Katopodes N.D. Model for flood propagation on initially dry land. J. Hydraul. Eng. 1987, 114:689-706.
2. Alcrudo F., Benkhaldoun F. Exact solutions to the Riemann problem of the shallow water equations with a bottom step. Comput. Fluids 2001, 30:643671.
3. Bernetti R., Titarev V.A., Toro E.F. Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry. J. Comput. Phys. 2008, 227:3212-3243.
4. Brufau P., Vázquez-Cendón M.E., Garci{dotless}́a-Navarro P. A numerical model for the flooding and drying of irregular domains. J. Numer. Methods Fluids 2002, 39:247-275.
5. Burguete J., Garci{dotless}́a-Navarro P. Efficient construction of high-resolution TVD conservative schemes for equations with source terms: application to shallow water flows. Int. J. Numer. Methods Fluids 2001, 37:209-248.
6. Courant R., Isaacson E., Rees M. On the solution of nonlinear hyperbolic differential equations by finite differences. Comm. Pure Appl. Maths. 1952, 5:243-255.
7. George D.L. Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation. J. Comput. Phys. 2008, 227:3089-3113.
8. Hubbard M.E., Garci{dotless}́a-Navarro P. Flux difference splitting and the balancing of source terms and flux gradients. J. Comput. Phys. 2000, 165:89-125.
9. Leveque R.J. Finite Volume Methods fot Hyperbolic Problems 2002, Cambridge University Press, New York, p. 311.
10. Liang Q.H., Marche F. Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour. 2009, 32:873-884.
11. Murillo J., Burguete J., Brufau P., Garci{dotless}́a-Navarro P. The influence of source terms on stability, accuracy and conservation in two-dimensional shallow flow simulation using triangular finite volumes. Int. J. Numer. Methods Fluids 2007, 54:543-590.
12. Murillo J., Garci{dotless}́a-Navarro P., Burguete J. Time step restrictions for well balanced shallow water solutions in non-zero velocity steady states. Int. J. Numer. Methods Fluids 2008, 56:661-686.
13. Murillo J., Garci{dotless}́a-Navarro P., Burguete J. Conservative numerical simulation of multicomponent transport in two-dimensional unsteady shallow water flow. J. Comput. Phys. 2009, 228:5539-5573.
14. Roe P.L. A basis for upwind differencing of the two-dimensional unsteady Euler equations. Numerical Methods in Fluid Dynamics 1986, vol. II. Oxford University Press, Oxford.
15. Rogers B.D., Borthwick A.G.L., Taylor P.H. Mathematical balancing of flux gradient and source terms prior to using Roes approximate Riemann solver. J. Comput. Phys. 2003, 168:422-451.
16. Rosatti G., Murillo J., Fraccarollo L. Generalized Roe schemes for 1D two-phase, free-surface flows over a mobile bed. J. Comput. Phys. 2007, 54:543-590.
17. Thacker W.C. Some exact solutions to the nonlinear shallow water equations. J. Fluid Mech. 1981, 107:499-508.
18. Toro E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics 1997, Springer, Berlin, p. 526.
19. Toro E.F. Shock-Capturing Methods for Free-Surface Shallow Flows 2001, Wiley, New York, p. 109.
20. Vázquez-Cendón M.E. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 1999, 148:497-498.
21. Zhou J.G., Causon D.M., Mingham C.G., Ingram D.M. The surface gradient method for the treatment of source terms in the shallow-water equations. J. Comput. Phys. 2001, 168:1-25.