Equations describing waves in tubes with elastic walls and numerical methods with low scheme dissipation

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.