A New Entropic Riemann Solver of Conservation Law of Mixed Type Including Ziti’s δ-Method with some Experimental Tests
Applied and Computational Mathematics
Volume 6, Issue 5, October 2017, Pages: 222-232
Received: Jun. 30, 2017;
Accepted: Jul. 11, 2017;
Published: Sep. 26, 2017
Views 2074 Downloads 142
Larbi Bsiss, Department of Mathematics, University Moulay Ismail, Meknes, Morocco
Cherif Ziti, Department of Mathematics, University Moulay Ismail, Meknes, Morocco
Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a new method called δ-ziti’s method for solving the governing equations. This method gives a new class of semi discrete, high-order scheme which are entropy conservative if the viscosity term is neglected. We implement a high resolution scheme for our mixed type problems that select the same viscosity solution as the Lax Friederich scheme with higher resolution. Several tests have been carried out to compare our results with those of   , in the same situations, we obtained the same results but faster thanks to the CFL condition which reaches 0.8 and the simplicity of the method. We consider three types of pressure in these tests: Cubic, Van der Waals and linear in pieces. The comparison proved that the δ-ziti's method respects the generalized Liu entropy conditions, e.g. the existence of a viscous profile.
A New Entropic Riemann Solver of Conservation Law of Mixed Type Including Ziti’s δ-Method with some Experimental Tests, Applied and Computational Mathematics.
Vol. 6, No. 5,
2017, pp. 222-232.
R. Abeyaratne and J. K. Knowles, “Kinetic relations and the propagation of phase boundaries in solids”, Arch Rational Mech. Anal. 114, 119 (1991).
R. Abeyaratne and J. K. Knowles, “Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries”, SIAM J. Appl. Math. 51, 1205 (1991).
M. Affouf and R. Caflisch, “A numerical study of Riemann problem solutions and stability for a system of viscous conservation laws of mixed type”, SIAM J. Appl. Math. 51, 605 (1991).
L. BSISS, C. ZITI, “A new numerical method for the integral approximation and solving the differential problems: Non-oscillating scheme, detecting the singularity in one and several dimensions”, J. Ponte, Vol. 73, Issue 2, pp. 126-172.
L. BSISS, C. ZITI, “A new Approximation (ziti’s δ-scheme) of the Entropic (Admissible) Solution of the Hyperbolic Problems in One and Several Dimensions: Applications to Convection, Burgers, Gas Dynamics and Some Biological Problems”, Turkish Journal of Analysis and Number Theory. 2016, 4(4), 98-108.
C. Chalons and P. G. Le Floch, “High-Order Entropy-Conservative and Kinetic Relations for van der Waals Fluids”, Journal of Computational Physics. (2001), 184-206.
C. Chalons and P. G. LeFloch, “A fully discrete scheme for diffusive-dispersive conservation laws”, Numerische Math. (2001), to appear.
B. Cockburn and H. Gau, “A model numerical scheme for the propagation of phase transitions in solids”, SIAM J. Sci. Comput. 17, 1092 (1996).
H. T. Fan, “A limiting “viscosity” approach to the Riemann problem for materials exhibiting a change of phase II”, Arch. Rational Mech. Anal. (1992) 116, 317.
H. Hattori, “The Riemann problem for a van der Waals fluid with the entropy rate admissibility criterion”, Arch. Rational Mech. Anal. 92, 247(1936).
B. T. Hayes and P. G. Le Floch, “Nonclassical shocks and kinetic relationsrelations: Scalar conservation laws”, Arch Rational Mech. Anal. 139, 1 (1997).
B. T. Hayes and P. G. Le Floch, “Nonclassical shocks and kinetic relations: Finite difference schemes”, SIAM J. Numer. Anal. 35, 2169 (1998).
B. T. Hayes and P. G. Le Floch, “Nonclassical shocks and kinetic relations: Strictly hyperbolic systems”, SIAMJ. Math. Anal. 31, 941 (2000).
D. Jacobs W. R. McKinney, and M. Shearer, “Traveling wave solutions of the modified Korteweg-de Vries Burgers equation”, J. Differential Equations 116, 448 (1995).
R. D. James, “The propagation of phase boundaries in elastic bars”, Arch. Rational Mech. Anal. 73, 125 (1980).
S. Jin, “Numerical integrations of systems of conservation laws of mixed type”, SIAM J. Appl. Math. 55, 1536 (1995).
P. G. Le Floch, “Propagating phase boundaries: Formulation of the problem and existence via the Glimm scheme”, Arch. Rational Mech. Anal. 123, 153 (1993).
P. G. Le Floch, “An introduction to nonclassical shocks of systems of conservation laws, in Proceedings of the International School on Theory and Numerics for Conservation Laws”, Freiburg Littenweiler (Germany), 20–24 October 1997, edited by D. Kr¨oner, M. Ohlberger, and C. Rohde, Lecture Notes in Computational Science and Engineering, (1998), p. 28.
P. G. LeFloch, “Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves”, E. T. H. Lecture Notes Series, 2001, to appear.
M. Shearer, “the Riemann problem for a class of conservation laws of mixed type”, J. Differential Equations 46, 426 (1982).
M. Shearer and Y. Yang, “The Riemann problem for the p-system of conservation laws of mixed type with a cubic nonlinearity”, Proc. Roy. Soc. Edinburgh. 125A, 675 (1995).
C.-W. Shu, “A numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting”, J. Comput. Phys. 100, 424 (1992).
M. Slemrod, “Admissibility criteria for propagating phase boundaries in a van der Waals fluid”, Arch. Rational Mech. Anal. 81, 301 (1983).
M. Slemrod and J. E. Flaherty, “Numerical integration of a Riemann problem for a van der Waals fluid, in Phase Transformations”, edited by C. A. Elias and G. John (Elsevier, Amsterdam, 1986).
L. Truskinovsky, “Dynamics of non-equilibrium phase boundaries in a heat conducting nonlinear elastic medium”, J. Appl. Math. Mech. 51, 777 (1987).
L. Truskinovsky, “Kinks versus shocks, in Shock Induced Transitions and Phase Structures in General Media”, edited by R. Fosdick, E. Dunn, and M. Slemrod, IMA Vol. Math. Appl. 52 (Springer- Verlag, New York 1993).