[Paper Review] Hodograph Method and Numerical Integration of Two Hyperbolic Quasilinear Equations. Part I. The Shallow Water Equations

E. V. Shiryaeva, M. Yu. Zhukov|arXiv (Cornell University)|Oct 10, 2014

Nonlinear Waves and Solitons6 references3 citations

TL;DR

This paper presents a hodograph-based numerical method for solving two hyperbolic quasilinear equations, specifically the shallow water equations, by transforming the original Cauchy problem into a system of ordinary differential equations (ODEs). The method leverages conservation laws and an explicit Riemann–Green function to enable accurate, non-approximate numerical integration, allowing for the construction of multi-valued solutions and capturing wave breaking phenomena with high precision.

ABSTRACT

In paper [S.I. Senashov, A. Yakhno. 2012. SIGMA. Vol.8. 071] the variant of the hodograph method based on the conservation laws for two hyperbolic quasilinear equations of the first order is described. Using these results we propose a method which allows to reduce the Cauchy problem for the two quasilinear PDE's to the Cauchy problem for ODE's. The proposed method is actually some similar method of characteristics for a system of two hyperbolic quasilinear equations. The method can be used effectively in all cases, when the linear hyperbolic equation in partial derivatives of the second order with variable coefficients, resulting from the application of the hodograph method, has an explicit expression for the Riemann-Green function. One of the method's features is the possibility to construct a multi-valued solutions. In this paper we present examples of method application for solving the classical shallow water equations.

Motivation & Objective

• To develop a numerical method for solving hyperbolic quasilinear PDEs without approximation, leveraging conservation laws.
• To enable the construction of multi-valued solutions, particularly for problems involving wave breaking.
• To provide an alternative to traditional finite difference or finite volume methods by reducing the PDE system to a Cauchy problem for ODEs.
• To validate the method on the classical shallow water equations with periodic initial conditions.
• To demonstrate the method’s effectiveness in capturing complex wave dynamics such as breaking and formation of droplets.

Proposed method

• The hodograph method is applied to two hyperbolic quasilinear first-order PDEs in Riemann invariants, using conservation laws to derive a second-order linear hyperbolic PDE.
• The method requires an explicit analytical expression for the Riemann–Green function of the derived second-order PDE to enable implicit solution representation.
• The original Cauchy problem for the PDEs is reformulated as a system of ODEs in the hodograph variables, which are then solved numerically.
• The transformation relies on characteristic speeds and flux-densities derived from conservation laws, with boundary conditions specified on characteristics.
• Numerical solution of the ODEs is performed using standard methods like Runge–Kutta, avoiding the need to solve nonlinear transcendental equations directly.
• The method is validated using periodic initial data for shallow water equations, with results visualized at multiple time steps.

Experimental results

Research questions

• RQ1 Can the hodograph method be effectively adapted to transform a system of hyperbolic quasilinear PDEs into a solvable ODE system for numerical integration?
• RQ2 How accurately can the method capture wave breaking and multi-valued solutions in shallow water flows?
• RQ3 What is the computational advantage of solving ODEs over solving nonlinear transcendental equations derived from implicit solutions?
• RQ4 Can the method be applied to other hyperbolic systems with known Riemann–Green functions, such as gas dynamics or soliton equations?
• RQ5 How does the method perform in simulating long-time evolution of periodic initial conditions in shallow water systems?

Key findings

• The method successfully transforms the shallow water equations into a Cauchy problem for ODEs, enabling high-precision numerical integration without approximation.
• Wave breaking is clearly observed in numerical results at t = 5.855, indicating the method’s capability to capture discontinuities.
• At t = 7.633 and t = 8.008, the formation of bubbles or droplets on the water surface is visually evident, suggesting the method captures complex free-surface dynamics.
• The use of hypergeometric and elliptic functions allows accurate evaluation of the Riemann–Green function, even for negative arguments via complex arithmetic or real-variable identities.
• The method outperforms direct solution of transcendental equations, as ODE solvers are more robust and efficient than Newton-type methods for such systems.
• The approach is generalizable to other systems with explicit Riemann–Green functions, including gas dynamics, soliton equations, and chromatography models.