An iterative scaling function procedure for solving scalar non-linear hyperbolic balance laws

TL;DR

This paper introduces an iterative scaling function method for solving scalar non-linear hyperbolic balance laws, demonstrating convergence and improved accuracy over traditional methods through theoretical analysis and numerical experiments.

Contribution

The paper presents a novel iterative procedure based on solution scaling for hyperbolic balance laws, with proven convergence and practical numerical implementation.

Findings

The iterative method converges in the $L^2$ framework for bounded, Lipschitz source terms.

Numerical results show improved accuracy over conventional first-order schemes.

The approach is feasible and effective for solving scalar non-linear hyperbolic balance laws.

Abstract

The scaling of the exact solution of a hyperbolic balance law generates a family of scaled problems in which the source term does not depend on the current solution. These problems are used to construct a sequence of solutions whose limiting function solves the original hyperbolic problem. Thus this gives rise to an iterative procedure. Its convergence is demonstrated both theoretically and analytically. The analytical demonstration is in terms of a local in time convergence and existence theorem in the $L^2$ framework for the class of problems in which the source term $s(q)$ is bounded, with $s(0) = 0$, is locally Lipschitz and belongs to $C^2(\mathbb{R}) \cap H^1 (\mathbb{R}) $. A convex flux function, which is usual for existence and uniqueness for conservation laws, is also needed. For the numerical demonstration, a set of model equations is solved, where a conservative finite volume method using a low-dissipation flux is implemented in the iteration stages. The error against reference solutions is computed and compared with the accuracy of a conventional first order approach in order to assess the gaining in accuracy of the present procedure. Regarding the accuracy only a first order scheme is explored because the development of a useful procedure is of interest in this work, high-order accurate methods should increase the computational cost of the global procedure. Numerical tests show that the present approach is a feasible method of solution.