Numerical Simulation of One-Dimensional Incompressible Transient Flow Using Characteristic Method with Midpoint Integration Formula

The Characteristic Method is a classical numerical simulation technique particularly suitable for solving one-dimensional incompressible transient flow problems. This method transforms partial differential equations into ordinary differential equations along characteristic lines, thereby simplifying the computational process. The Midpoint Formula is employed for numerical integration along these characteristic lines to enhance calculation accuracy. In one-dimensional incompressible transient flow simulations, the method typically involves mass and momentum conservation equations. The core concept of the Characteristic Method utilizes flow characteristic lines (i.e., paths of information propagation) to convert partial differential equations into ordinary differential equations along these trajectories. This approach not only captures the transient behavior of fluids but also efficiently handles wave propagation phenomena. The Midpoint Formula plays a crucial role in the Characteristic Method by discretizing characteristic lines and applying midpoint approximation for integral calculations at each time step. Compared to basic Euler methods, this technique offers superior accuracy and better simulates nonlinear effects and shock wave phenomena in fluid flow. Implementation typically involves: - Discretizing the spatial domain into nodes with Δx spacing - Calculating characteristic directions using flow velocity and wave celerity - Applying midpoint integration with time step Δt to update flow variables This numerical simulation method finds widespread applications in pipeline flow analysis, water hammer effect studies, and hydraulic system transient process investigations. Its advantages include high computational efficiency and accurate capturing of key physical phenomena in fluid dynamics. Code implementation often features: - Matrix operations for solving characteristic equations - Iterative algorithms for handling boundary conditions - Stability checks using Courant–Friedrichs–Lewy (CFL) conditions - Visualization routines for wave propagation patterns