The accurate capturing of shock waves by numerical methods has long been a focus of attention in engineering owing to singularity problems in discontinuities. In this article, a novel coupled Euler–Lagrange method (CELM) is proposed to capture shock waves and discontinuities with high resolution and high order of mapping accuracy. CELM arranges the Lagrange particles on an Euler grid to track the discontinuous points automatically, and the data pertaining to the grids and particles interact via a weighted mutual mapping method that not only achieves fourth-order accuracy in a smooth area of the solution but also maintains a steep discontinuous transition in the discontinuous area. In the virtual particle method, virtual particles are derived from the existing real particles; thus, the inflow and outflow of the particles and interpolation accuracy of the boundary are more easily realized. An accuracy test and energy convergence test demonstrated the fourth-order convergence accuracy and low energy dissipation of the CELM; the method exhibited lower error and better conservation ability than high-precision schemes such as WENO3 and WENO5. The Sod shock tube problem and Woodward–Colella problem showed higher discontinuity resolution of the CELM and ability to accurately track discontinuity points. Examples of Riemann problems were employed to prove that the CELM exhibits lower dissipation and higher shock resolution than WENO3 and WENO5. The CELM also showed an accurate structure based on particle distribution. Shockwave diffraction tests were conducted to prove that the CELM results showed good agreement with the experimental data and exhibited an accurate expansion wave. The CELM can also accurately simulate the collision of an expansion wave and vortex.

中文翻译：

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一种新颖的耦合欧拉-拉格朗日方法，用于高分辨率冲击和不连续性捕获

由于不连续性的奇点问题，用数值方法精确捕捉冲击波长期以来一直是工程界关注的焦点。在本文中，提出了一种新颖的耦合欧拉-拉格朗日方法（CELM），以高分辨率和高阶映射精度捕获冲击波和不连续性。CELM将拉格朗日粒子排列在欧拉网格上，自动跟踪不连续点，网格和粒子的数据通过加权相互映射方法进行交互，不仅在解的平滑区域达到四阶精度，而且保持不连续区域中的陡峭不连续过渡。在虚拟粒子法中，虚拟粒子是由现有的真实粒子推导出来的；从而更容易实现粒子的流入和流出以及边界的插值精度。精度测试和能量收敛测试证明了CELM的四阶收敛精度和低能量耗散；该方法比WENO3和WENO5等高精度方案表现出更低的误差和更好的守恒能力。Sod 激波管问题和 Woodward-Colella 问题显示了 CELM 更高的不连续性分辨率以及准确跟踪不连续点的能力。利用黎曼问题的例子证明了CELM比WENO3和WENO5具有更低的耗散和更高的冲击分辨率。CELM 还显示了基于颗粒分布的精确结构。冲击波衍射测试证明CELM结果与实验数据吻合良好，并表现出准确的膨胀波。CELM还可以准确模拟膨胀波和涡流的碰撞。