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A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. (English) Zbl 1055.65104
The author proves the convergence of a simple difference scheme based on the Engquist-Osher numerical flux for the following Cauchy problem: $u_t + (k(x)f(u)-a(x))_x = 0,\quad u(x,0)=u_0(x),$ with $$(x,t)\in {\mathbb R}\times {\mathbb R}^+$$. It is worth noting that the author uses discretized versions of $$k$$ and $$a$$ that are staggered with respect to that of $$u$$, resulting in a reduction in complexity. The assumptions concerning the data are that $$u_0\in L^1\cap L^\infty\cap BV$$ and $$a,k\in L^1_{loc}\cap L^\infty\cap BV$$ (moreover, $$k$$ is supposed to be bounded from below by a positive constant), being $$BV$$ the space of locally integrable functions having bounded total variation. In particular, the flux $$k(x)f(u)-a(x)$$ has a possibly discontinuous spatial dependence through the positive coefficient $$k(x)$$ and the source term $$a(x)$$.
This is an extension of the previous work by the author [SIAM J. Numer. Anal. 38, No. 2, 681–698 (2000; Zbl 0972.65060)], where the flux was strictly concave and had no source term. In the present paper, the convergence is proved for a large class of nonconvex flux functions. Apart from other assumptions on $$f$$, not easy to state briefly (see Proposition 2.1 and Theorem 3.1 in the paper), the one substituting concavity is that $$f$$ has finitely many critical points.
The main result (Theorem 3.1) is a Lax-Wendroff type theorem. It proves simultaneously that the Cauchy problem has a weak solution and that the sequence generated by the difference scheme contains a subsequence converging in $$L^1_{loc}({\mathbb R}\times {\mathbb R}^+)$$ to that weak solution. Uniqueness and entropy satisfaction are potential areas for further investigation.
Other questions treated in this paper are: monotonicity of the scheme, a priori bounds on the numerical approximations, maximum principles and variation stability for the flux $$kf(u)-a$$, and some remarks on numerical experience with the algorithm. From a practical point of view, the scheme is limited to be at most of first order accuracy due to the monotonicity, while its main value is simplicity. In fact, given a working computer program implementing a simple upwind difference scheme for the conservation law $$u_t + f(u)_x = 0$$, only minor modifications are required to produce a program implementing the present scheme for the significantly harder Cauchy problem studied in this paper.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35A35 Theoretical approximation in context of PDEs
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