A method of fractional steps for scalar conservation laws without the CFL condition
Authors:
Helge Holden and Nils Henrik Risebro
Journal:
Math. Comp. 60 (1993), 221232
MSC:
Primary 65M12; Secondary 35L65
MathSciNet review:
1153165
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We present a numerical method for the ndimensional initial value problem for the scalar conservation law . Our method is based on the use of dimensional splitting and Dafermos's method to solve the onedimensional equations. This method is unconditionally stable in the sense that the time step is not limited by the space discretization. Furthermore, we show that this method produces a subsequence which converges to the weak entropy solution as both the time and space discretization go to zero. Finally, two numerical examples are discussed.
 [1]
Frode
Bratvedt, Kyrre
Bratvedt, Christian
F. Buchholz et al., Front tracking for petroleum reservoirs,
applications (Oslo, 1988) Cambridge Univ. Press, Cambridge, 1992,
pp. 409–427. MR 1190515
(93h:76071)
 [2]
Edward
Conway and Joel
Smoller, Clobal solutions of the Cauchy problem for quasilinear
firstorder equations in several space variables, Comm. Pure Appl.
Math. 19 (1966), 95–105. MR 0192161
(33 #388)
 [3]
Michael
Crandall and Andrew
Majda, The method of fractional steps for conservation laws,
Numer. Math. 34 (1980), no. 3, 285–314. MR 571291
(81j:65101), http://dx.doi.org/10.1007/BF01396704
 [4]
Constantine
M. Dafermos, Polygonal approximations of solutions of the initial
value problem for a conservation law, J. Math. Anal. Appl.
38 (1972), 33–41. MR 0303068
(46 #2210)
 [5]
S. K. Godunov, Finite difference methods for numerical computations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), 271306 (Russian).
 [6]
Helge
Holden and Lars
Holden, On scalar conservation laws in one dimension,
applications (Oslo, 1988) Cambridge Univ. Press, Cambridge, 1992,
pp. 480–509. MR 1190518
(93i:65101)
 [7]
H.
Holden, L.
Holden, and R.
HøeghKrohn, A numerical method for first order nonlinear
scalar conservation laws in one dimension, Comput. Math. Appl.
15 (1988), no. 68, 595–602. Hyperbolic partial
differential equations. V. MR 953567
(90c:65112), http://dx.doi.org/10.1016/08981221(88)902829
 [8]
S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSRSb. 10 (1970), 217243.
 [9]
N. Kuznetsov, Weak solution of the Cauchy problem for a multidimensional quasilinear equation, Math. Notes 2 (1967), 733739.
 [10]
A.
I. Vol′pert, Spaces 𝐵𝑉 and quasilinear
equations, Mat. Sb. (N.S.) 73 (115) (1967),
255–302 (Russian). MR 0216338
(35 #7172)
 [1]
 F. Bratvedt, K. Bratvedt, C. Buchholz, T. Gimse, H. Holden, L. Holden, and N. H. Risebro, Front tracking for petroleum reservoirs, in Ideas and Methods in Mathematics and Physics (S. Albeverio, J. E. Fenstad, H. Holden, T. Lindstrøm, eds.), Cambridge Univ. Press, Cambridge, 1992, pp. 409427. MR 1190515 (93h:76071)
 [2]
 E. Conway and J. Smoller, Global solutions of the Cauchy problem for quasilinear first order equations in several space variables, Comm. Pure Appl. Math. 19 (1966), 95105. MR 0192161 (33:388)
 [3]
 M. Crandall and A. Majda, The method of fractional steps for conservation laws, Numer. Math. 34 (1980), 285314. MR 571291 (81j:65101)
 [4]
 C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 3341. MR 0303068 (46:2210)
 [5]
 S. K. Godunov, Finite difference methods for numerical computations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), 271306 (Russian).
 [6]
 H. Holden and L. Holden, On scalar conservation laws in one dimension, in Ideas and Methods in Mathematics and Physics (S. Albeverio, J.E. Fenstad, H. Holden, T. Lindstrøm, eds.), Cambridge Univ. Press, Cambridge, 1992, pp. 480509. MR 1190518 (93i:65101)
 [7]
 H. Holden, L. Holden, and R. HøeghKrohn, A numerical method for first order nonlinear scalar conservation laws in one dimension, Comput. Math. Appl. 15 (1988), 595602. MR 953567 (90c:65112)
 [8]
 S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSRSb. 10 (1970), 217243.
 [9]
 N. Kuznetsov, Weak solution of the Cauchy problem for a multidimensional quasilinear equation, Math. Notes 2 (1967), 733739.
 [10]
 A. I. Vol'pert, The spaces BV and quasilinear equations, Math. USSRSb. 2 (1967), 225267. MR 0216338 (35:7172)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65M12,
35L65
Retrieve articles in all journals
with MSC:
65M12,
35L65
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199311531655
PII:
S 00255718(1993)11531655
Keywords:
Dimensional splitting,
scalar conservation law,
fractional steps,
numerical methods
Article copyright:
© Copyright 1993 American Mathematical Society
