论文标题
一种迭代缩放函数程序，用于求解标量非线性双曲平衡定律
An iterative scaling function procedure for solving scalar non-linear hyperbolic balance laws
论文作者
论文摘要
双曲线平衡定律的精确解决方案的缩放产生了一个缩放问题家族，其中源项不取决于当前解决方案。这些问题用于构建一系列解决方案的序列，该解决方案的限制函数解决了原始双曲线问题。因此，这引起了迭代程序。理论和分析都证明了它的收敛性。分析性演示是根据$ l^2 $框架中的本地时间收敛和存在定理来介绍的，其中源术语$ s（q）$有限，$ s（0）= 0 $是本地lipschitz，属于Lipschitz，属于$ c^2（\ cap {r}）\ cap h^1（\ cap h^1（\ MathBB）还需要一个凸通量函数，这通常是生存和保护法的独特性。对于数值演示，解决了一组模型方程，其中使用低径流通量的保守有限体积方法在迭代阶段实现。计算针对参考解决方案的误差，并将其与常规一阶方法的准确性进行了比较，以评估本程序准确性的获得。关于准确性仅探讨了一阶方案，因为有用的过程的开发在这项工作中引起了人们的关注，高阶准确方法应增加全球过程的计算成本。数值测试表明，目前的方法是一种可行的解决方案方法。
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.
