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Numerical methods for weak solution of wave equation with van der Pol type nonlinear boundary conditions
Author(s) -
Liu Jun,
Huang Yu,
Sun Haiwei,
Xiao Mingqing
Publication year - 2016
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21997
Subject(s) - mathematics , riemann hypothesis , nonlinear system , type (biology) , boundary value problem , mathematical analysis , degree (music) , boundary (topology) , van der pol oscillator , partial differential equation , riemann problem , reflection (computer programming) , scheme (mathematics) , wave equation , order (exchange) , numerical analysis , computer science , ecology , physics , quantum mechanics , acoustics , biology , programming language , finance , economics
We develop computational methods for solving wave equation with van der Pol type nonlinear boundary conditions under the framework of weak solutions. Based on the wave reflection on the boundaries, we first solve the Riemann invariants by constructing two iteration mappings, and then show that the weak solution can be obtained by the integration of the Riemann invariants on the boundaries. If the compatible conditions are not satisfied or only hold with a low degree, a high‐order integration method is developed for the numerical solution. When the initial condition is sufficiently smooth and compatible conditions hold with a sufficient degree, we establish a sixth‐order finite difference scheme, which only needs to solve a linear system at any given time instance. Numerical experiments are provided to demonstrate the effectiveness of the proposed approaches. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 373–398, 2016