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SYMPOSIUM MINISYMPOSIA

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FIRST SOUTHERN SYMPOSIUM ON COMPUTING

December 4-5, 1998
University of Southern Mississippi
Hattiesburg, Mississippi


ABSTRACT

Entropy Consistent, Implicit TVD Methods with High Accuracy for Conservation Law

Xuefeng Li

Implicit methods for solving hyperbolic conservation laws are developed in this paper. They are proven to be total variation diminishing(TVD) and convergent to weak solutions which also satisfy the entropy condition (Entropy Consistency) for the case of scalar hyperbolic conservation laws. Numerical experiments demonstrated that these methods can also be applied to system of hyperbolic conservation laws without much modification. And the numerical solutions produced by those methods are comparable to those produced by explicit higher order accurate Godunov-type methods.

The paper is a sequel to the author's previous work [Li, 1998] where a class of highly accurate numerical methods was developed which are TVD and entropy consistent in the case of scalar hyperbolic conservation laws. Often in an explicit method the numerical time step is restricted due to stability considerations, and an usual way to overcome the restriction is to use an appropriate implicit method. This is indeed the focus of this paper though it is pretty much self-contained.

In recent years, many implicit numerical methods [Yee, Warming & Harten, 1985; Yee, 1987; Blunt & Rubin, 1992; Wu, Y.H. & Wu H.M, 1992; Wilcoxson & Manousiouthakis, 1994] were developed for solving hyperbolic conservation laws:

t u +x f(u)
= 0,       (x,t) Î (-¥,+¥)×(t0,+¥),
(1.1a)
u(x,t0)
= u0(x),       x Î (-¥,+¥),
(1.1b)
where a solution vector u has m components and is a function of two variables x and t. That is, u = u(x,t) = (u1,¼,um)T, f(u) = (f1(u),¼,fm(u))T, and matrix uf has m distinct real eigenvalues. When m = 1, (1.1) represents a scalar conservation law; whereas when m > 1, (1.1) represents a system of conservation laws which produces quite a different scenario for any detailed analysis of numerical methods for (1.1).

The goal of this paper is to develop a new class of implicit numerical methods for solving (1.1) which are proven to produce numerical solutions convergent to physically relevant solutions in the case when m = 1.

References

  1. Blunt, M. and Rubin, B., (1992). Implicit flux limiting schemes for petroleum reservoir simulation, J. Comput. Phys., 102, no.1, 194-210.

  2. Li, X. (1998). Achieving High Accuracy using Piecewise Constant Functions in Conservation Laws, to appear in the Electronic Journal of Differential Equations. 0

  3. Wilcoxson, M. and Manousiouthakis, V., (1994). On an implicit ENO scheme, J. Comput. Phys., 115, no.2, 376-389.

  4. Wu, Y. H. and Wu, H. M., (1992). The entropy condition for implicit TVD schemes, J. Comput. Math., 10, no.2, 155-166.

  5. Yee, H. C., (1987). Construction of explicit and implicit symmetric TVD schemes and their applications, J. Comput. Phys., 68, no.1, 151-179.

  6. Yee, H. C., Warming, R. F. and Harten, A., (1985). Implicit total variation diminishing (TVD) schemes for steady-state calculations, J. Comput. Phys., 57, no.3, 327-360.


Getting More Information

To obtain more information about the meeting send e-mail to: fscc98@pax.st.usm.edu.

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