Abstract

Equations for a tube with elastic waves (tube with controlled pressure, fluid-filled tube, and gas-filled tube) are considered. A full membrane model and the nonlinear theory of hyperelastic materials are used for describing the tube walls. The Riemann problem is solved, and its solutions confirm the theory of reversible discontinuity structures. Dispersion of short waves for such equations vanishes; for this reason, dissipative discontinuity structures can be included. The equations under examination are complicated due to which general numerical methods are developed. Application of the centered three-layer cross-type scheme and schemes based on the approximation of time derivatives using the Runge–Kutta method of various orders is considered. A technology for correcting schemes based on the Runge–Kutta method by adding dissipative terms is developed. In the scalar case, the third- and fourth-order methods do not require correction. The possibility of using terms with high-order derivatives for computing solutions that simultaneously include dissipative and nondissipative discontinuities is analyzed.

中文翻译：

---

弹性壁管中描述波的方程和低耗散数值方法

摘要

考虑具有弹性波的管（压力受控的管，流体填充的管和气体填充的管）的方程式。使用全膜模型和超弹性材料的非线性理论来描述管壁。解决了黎曼问题，其解决方案证实了可逆不连续结构的理论。这种方程式的短波色散消失了。因此，可以包括耗散的不连续结构。由于要开发通用的数值方法，因此所研究的方程很复杂。考虑了中心三层交叉类型方案以及基于各种阶数的Runge-Kutta方法基于时间导数近似的方案的应用。开发了一种基于Runge-Kutta方法并通过添加耗散项来校正方案的技术。在标量情况下，三阶和四阶方法不需要校正。分析了将具有高阶导数的项用于同时包含耗散和非耗散不连续性的计算解决方案的可能性。