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A semi‐discrete convergent scheme for a quasilinear hyperbolic equation
Author(s) -
Kannan R.,
Ortega R.
Publication year - 1987
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690030406
Subject(s) - mathematics , scheme (mathematics) , continuation , convergence (economics) , hyperbolic partial differential equation , type (biology) , weak solution , mathematical analysis , term (time) , pure mathematics , partial differential equation , computer science , ecology , physics , quantum mechanics , economics , biology , programming language , economic growth
We establish here the convergence (thereby proving the existence) of a semi‐discrete scheme for the quasilinear hyperbolic equationwhere x ∈ R n , t ∈ [0, T ], and ϕ ∈ L ∞ ( R n ). It is well known that the above problem does not necessarily have global classical solutions and the usual concepts of weak solution. do not lead to a unique solution The existence of a unique solution to the above problem in a suitable sense was established in [3], where a parabolic problem obtained by introducing the term −ϵΔ u was studied and then the behavior as ϵ → 0 was discussed. A difference scheme approach to a problem of the above type where ϕ i does not depend on x and t and Ψ does not depend on u was also studied in [2]. The aim of this paper is to present a proof for the case when ϕ depends on x , Ψ depends on u , and the technical complications in this case are nontrivial. The discussions in this paper my be considered as continuation of the ideas in the above papers.