In the present work, a new numerical strategy is designed to approximate the Riemann solutions of systems of conservation laws. Here, the main difficulty comes from the definition of the discontinuous solutions. Indeed, the shock solutions are no longer selected by entropy criterion but they are defined as the zero limit of a diffusive–dispersive system. As a consequence, the solutions of interest may contain non classical shocks. In order to derive a suitable numerical approach, the Dafermos diffusion technique is adopted here. Then, the PDE initial value problem is reformulated as an ODE boundary value problem. A fourth-order finite difference scheme is introduced to approximate the solution of this ODE boundary value problem. In this work, a particular attention is paid on the existence of discrete solutions and several numerical experiments illustrate the relevance of the derived numerical strategy.

中文翻译：

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基于Dafermos重构类型问题的零扩散-色散极限的Riemann解近似

在目前的工作中，设计了一种新的数值策略来近似守恒律系统的黎曼解。这里，主要困难来自不连续解的定义。的确，激波解不再由熵准则选择，而是定义为扩散-色散系统的零极限。结果，感兴趣的解决方案可能包含非经典冲击。为了得出合适的数值方法，此处采用了Dafermos扩散技术。然后，将PDE初始值问题重新表述为ODE边界值问题。引入了四阶有限差分方案来近似求解该ODE边值问题的解。在这项工作中