稀疏和激波曲线中的共振：局部分析和延拓方法的数值计算
Resonance in rarefaction and shock curves: local analysis and numerics of the continuation method
摘要：
在本文中，我们描述了某些关键步骤的发展算法寻找黎曼解决方案的系统的守恒律。我们放松了经典的假设严格双曲性和真正的非线性Lax。首先，我们提出了一个程序，连续波曲线超过点的特征速度一致，即，在最大余维的波形曲线点处。这个过程要求重合轨迹的两侧都是严格双曲性。利用广义Jordan链正则化了严格双曲性的损失，构造了一个四重子流形结构，使波曲线在其上连续。其次，我们分析了真正的非线性损失的情况。我们证明了一个新的结果：当复合波穿过非局部复合波曲线的拐点轨迹或异常部分时，复合波曲线的存在。在这个意义上说，我们找到条件下的复合领域是很好的定义和它的奇点可以被删除，允许使用我们的延续方法。最后，我们给出了一个非严格双曲型守恒律组的数值例子。
英文摘要：
In this paper, we describe certain crucial steps in the development of an algorithm for finding the Riemann solution in systems of conservation laws. We relax the classical hypotheses of strict hyperbolicity and genuine nonlinearity of Lax. First, we present a procedure for continuing wave curves beyond points where characteristic speeds coincide, i.e., at wave curve points of maximal codimensionality. This procedure requires strict hyperbolicity on both sides of the coincidence locus. Loss of strict hyperbolicity is regularized a Generalized Jordan Chain, which serves to construct a four-fold submanifold structure on which wave curves can be continued. Second, we analyze the case of loss of genuine nonlinearity. We prove a new result: the existence of composite wave curves when the composite wave traverses either the inflection locus or an anomalous part of the non-local composite wave curve. In this sense, we find conditions under which the composite field is well defined and its singularities can be removed, allowing use of our continuation method. Finally, we present numerical examples for a non-strictly hyperbolic system of conservation laws.
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