What are particle based methods to solve the wave equation and allied PDEs?
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Particle methods offer a powerful approach to solving wave equations and related partial differential equations (PDEs). These methods involve representing the solution using a collection of particles located at specific points, each carrying certain weights . By evolving the positions and weights of these particles over time based on governing equations, particle methods can accurately capture nonlinear waves with high resolution. They excel in scenarios like wave breaking and runup analysis, showcasing robustness and precision. Additionally, advancements like the Homotopy Perturbation Method (HPM) have been applied to wave equations with non-local conditions, demonstrating accurate solutions with faster convergence and reduced computational effort. These methods eliminate the need for a recovery step in computations, simplifying the process and enhancing accuracy.
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2 Citations | Particle methods for the semiclassical limit of the Schrödinger equation and Vlasov-Poisson equations utilize conservation of charge to construct density directly, avoiding the need for a recovery step. |
| Not addressed in the paper. | |
Open access•Journal Article 25 Citations | Particle methods, like Lagrangian Particle Method, solve Navier-Stokes equations without a grid, ideal for analyzing wave breaking and runup processes, offering robustness and precision in numerical wave flumes. |
19 Citations | Particle methods approximate PDE solutions using Dirac delta-functions. They excel in capturing nonlinear waves with low numerical diffusion, extending to convection-diffusion and dispersive equations, and general nonlinear problems. |
| Particle methods represent solutions of PDEs using groups of particles with masses and positions, evolving over time. They are effective for approximating solutions of dispersive equations in various fields. |
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