DOI: https://doi.org/10.26089/NumMet.v17r437

Implicit and time reversible CABARET schemes for quasilinear shallow water equations

Authors

  • V.M. Goloviznin
  • D.Yu. Gorbachev
  • A.M. Kolokolnikov
  • P.A. Maiorov
  • B.A. Tlepsuk

Keywords:

CABARET scheme
shallow water equations
conservative schemes
time reversible schemes
numerical simulation

Abstract

A new implicit unconditionally stable scheme for the one-dimensional shallow water equations is proposed. This implicit scheme retains all the features of the explicit CABARET (Compact Accurately Boundary Adjusting-REsolution Technique) difference scheme. Dissipative and dispersion properties of this new scheme are analyzed; an algorithm of its numerical solution is discussed. Some examples of solving the Riemann problem are considered.

New computational technologies

Downloads

Published

2016-09-27

Issue

Vol. 17 (2016): Issue 4.

Section

Section 1. Numerical methods and applications

Author Biographies

V.M. Goloviznin

Nuclear Safety Institute (IBRAE) of RAS
• Head of Department

D.Yu. Gorbachev

Lomonosov Moscow State University
• Student

A.M. Kolokolnikov

Lomonosov Moscow State University
• Student

P.A. Maiorov

Lomonosov Moscow State University
• Student

B.A. Tlepsuk

Lomonosov Moscow State University
• Student

References

  1. B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978; Amer. Math. Soc., Providence, 1982).
  2. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; CRC Press, Boca Raton, 2001).
  3. A. I. Zhukov, “Application of the Method of Characteristics to the Numerical Solution of One-Dimensional Problems in Gas Dynamics,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 58 (3), 3-150 (1960).
  4. K. M. Magomedov and A. S. Kholodov, Grid-Characteristic Numerical Methods (Nauka, Moscow, 1988) [in Russian].
  5. V. M. Goloviznin, S. A. Karabasov, and I. M. Kobrinskii, “Balance-Characteristic Schemes with Separated Conservative and Flux Variables,” Mat. Model. 15 (9), 29-48 (2003).
  6. V. M. Goloviznin, M. A. Zaitsev, S. A. Karabasov, and I. A. Korotkin, New CFD Algorithms for Multiprocessor Computer Systems (Mosk. Gos. Univ., Moscow, 2013) [in Russian].
  7. A. A. Samarskii and Yu. P. Popov, Difference Schemes for Solving Gas Dynamics Problems (Nauka, Moscow, 1992) [in Russian].
  8. V. M. Goloviznin and A. A. Samarskii, “Finite Difference Approximation of Convective Transport Equation with Space Splitting Time Derivative,” Mat. Model. 10 (1), 86-100 (1998).
  9. V. M. Goloviznin and A. A. Samarskii, “Some Characteristics of Finite Difference Scheme ’cabaret’,” Mat. Model. 10 (1), 101-116 (1998).
  10. V. M. Goloviznin, V. A. Isakov, and A. V. Solovjev, “An Approximation Algorithm for Treatment of Sound Points in the CABARET Scheme,” Differ. Uravn. 52 (2016) (in press).
  11. V. M. Goloviznin and S. A. Karabasov, “Nonlinear Correction of Cabaret Scheme,” Mat. Model. 10 (12), 107-123 (1998).