Rope: Implicit Solver PDE Simulation of Moving Pulse

Similar to the previous rope simulation, but this time using an implicit solver instead of an explicit one. Even after 100,000 time steps, the initial pulse remains coherent. This shows why implicit solvers are essential for hyperbolic PDEs (such as the wave equation). Intuitively, the implicit scheme has greater stability because it encodes global information, while the explicit scheme just uses local information. In the explicit scheme, the acceleration at a given point is fully determined by the positions of neighboring points at the previous time step. In contrast, the implicit scheme has velocities at the current time step determined by a matrix equation Mv = u, where M is tridiagonal and u is determined by the state at the previous time step. The matrix equation introduces global dependencies among the new velocities.

Category:

Science & Technology

Tags:

• PDE
• simulation
• math
• partial
• differential
• equation
• implicit

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PDEs are sweet.