Abstract
In this paper, we mainly study an initial and boundary value problem of a two-dimensional fourth-order hyperbolic equation. Firstly, the fourth-order equation is written as a system of two second-order equations by introducing two new variables. Next, in order to design an implicit compact finite difference scheme for the problem, we apply the compact finite difference operators to obtain a fourth-order discretization for the second-order spatial derivatives and the Crank–Nicolson difference scheme to obtain a second-order discretization for the first-order time derivative. We prove the unconditional stability of the scheme by the Fourier method. Then a convergence analysis is given by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.
Highlights
IntroductionThe two-dimensional fourth-order hyperbolic equations have very important physical background and a wide range of applications
Let Ω = (0, a) × (0, b) and we consider the two-dimensional fourth-order hyperbolic equation with initial and boundary conditions: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨(((bca))) ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩(d) utt + ρ 2u = f (x, y, t), (x, y, t) ∈ Ω × (0, T], u(x, y, 0) = f1(x, y), ∂u ∂t |(x,y,0) = f2(x, y),(x, y) ∈ Ω, u|x=0 = h1(y, t), u|x=a = h2(y, t), u|y=0 = h3(x, t), u|y=b = h4(x, t), t ∈ [0, T], u|x=0 = g1(y, t), u|x=a = g2(y, t), u|y=0 = g3(x, t), u|y=b = g4(x, t), t ∈ [0, T], (1) where utt ∂2u ∂t2
We prove the stability for the high-order compact difference scheme by the Fourier method
Summary
The two-dimensional fourth-order hyperbolic equations have very important physical background and a wide range of applications. Compared with the second-order equations [6,7,8,9,10,11], it is usually necessary to use higher-order finite element methods or thirteen-point difference schemes in order to solve the numerical solution of the two-dimensional fourth-order equations. The convergence of the high-order compact difference scheme is given by the energy method.
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