双曲守恒律的稀疏波追踪方法
A rarefaction-tracking method for hyperbolic conservation laws
AI总结：
提出一种基于局部相似解和粒子追踪的数值方法，用于一维标量守恒律，通过稀疏波和压缩波近似解，并保持无耗散特性。
摘要：
我们提出了一种一维空间中标量守恒律的数值方法。解通过局部相似解来近似。虽然许多常用方法基于激波，但本文提出的方法使用稀疏波和压缩波。解由携带函数值并根据特征线法移动的粒子表示。在两个相邻粒子之间，通过守恒律的解析相似解定义插值。粒子的相互作用代表特征线的碰撞。通过合并粒子以保持函数下的总面积守恒，从而解析产生的激波。该方法具有变差减小的特性，然而在激波附近没有数值耗散。尽管激波没有被显式追踪，但它们可以被准确定位。我们给出了数值例子，并概述了该方法的具体应用和扩展。
英文摘要：
We present a numerical method for scalar conservation laws in one space dimension. The solution is approximated by local similarity solutions. While many commonly used approaches are based on shocks, the presented method uses rarefaction and compression waves. The solution is represented by particles that carry function values and move according to the method of characteristics. Between two neighboring particles, an interpolation is defined by an analytical similarity solution of the conservation law. An interaction of particles represents a collision of characteristics. The resulting shock is resolved by merging particles so that the total area under the function is conserved. The method is variation diminishing, nevertheless, it has no numerical dissipation away from shocks. Although shocks are not explicitly tracked, they can be located accurately. We present numerical examples, and outline specific applications and extensions of the approach.
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