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.

中文翻译：

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求解标量非线性双曲平衡定律的迭代标度函数程序

双曲平衡定律的精确解的标度会产生一系列标度问题，其中源项不依赖于当前解。这些问题用于构造一系列解，其极限函数解决原始双曲问题。因此，这产生了迭代过程。它的收敛性在理论上和分析上都得到了证明。分析证明是根据 $L^2$ 框架中的局部时间收敛性和存在定理，用于源项 $s(q)$ 有界的问题类别，$s(0) = 0 $, 是局部 Lipschitz 并且属于 $C^2(\mathbb{R}) \cap H^1 (\mathbb{R}) $。还需要一个凸通量函数，它通常用于守恒定律的存在性和唯一性。对于数值演示，求解一组模型方程，其中在迭代阶段实施使用低耗散通量的保守有限体积方法。计算相对于参考解的误差，并与传统一阶方法的精度进行比较，以评估当前程序的精度增益。关于准确性，仅探索一阶方案，因为这项工作对有用程序的开发感兴趣，高阶准确方法应增加全局程序的计算成本。数值试验表明，该方法是一种可行的求解方法。计算相对于参考解的误差，并与传统一阶方法的精度进行比较，以评估当前程序的精度增益。关于准确性，仅探索一阶方案，因为这项工作对有用程序的开发感兴趣，高阶准确方法应增加全局程序的计算成本。数值试验表明，该方法是一种可行的求解方法。计算相对于参考解的误差，并与传统一阶方法的精度进行比较，以评估当前程序的精度增益。关于准确性，仅探索一阶方案，因为这项工作对有用程序的开发感兴趣，高阶准确方法应增加全局程序的计算成本。数值试验表明，该方法是一种可行的求解方法。