A difference scheme for conservation laws with a discontinuous flux: the nonconvex case.

*(English)*Zbl 1055.65104The 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.

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 |