Numerical solution of linear wave equation with strong dissipative term

doi:10.1016/j.amc.2003.10.045

Applied Mathematics and Computation

Volume 159, Issue 2, 6 December 2004, Pages 341-348

https://doi.org/10.1016/j.amc.2003.10.045 Get rights and content

Abstract

In this study, for the periodic problem with respect to time of wave theory, three level difference scheme is presented. The difference scheme is constructed by the method of integral identities with the use of linear basis function and interpolating quadrature rules with the remainder terms in integral form. Error of difference solution is estimated. The theoretical results are controlled on a numerical example.

Introduction

We discuss the following boundary value problem $Lu≡ ∂^{2} u ∂ t^{2} +L_{1} ∂ u ∂ t +L_{0} u=f(x,t), (x,t)∈D=(0,ℓ)x(0,T],$ $u(0,t)=u(ℓ,t)=0,$ $u(x,0)=u(x,T),$ $∂ u ∂ t (x,0)= ∂ u ∂ t (x,T),$ where $L_{1} ∂ u ∂ t =− ∂ ∂ x a(x,t) ∂^{2} u ∂ t ∂ x +b(x,t) ∂ u ∂ t,$ $L_{0} u=− ∂ ∂ x c(x,t) ∂ u ∂ x +d(x,t)u$ and a, b, c, d, f are sufficiently smooth functions in $D$ .

The equations of this type arise in many areas of mathemetical physics and fluid mechanics. These are used for studies about communication lines, electron plasm waves in plasmas, ion acoustics waves and other physical models [4], [5], [9].

We presented three-level difference scheme for this problem. For wave equation with strong dissipative term, difference schemes is constructed, mathematical researches are done and approximate error is presented that the convergence is O(h²+τ²), has been proved.

Existence, uniqueness and stability of exact solution of this type problems were investigated by several mathematician [6], [7], [8], [10].

And also, the numerical solutions of this type equations in simpler models are researched [1], [2], [3].

Section snippets

Establish of difference scheme

We suppose that $0<α⩽a(x,t)⩽a^{∗}$ , $0⩽b_{∗} ⩽b(x,t)⩽b^{∗}$ , $0<c_{∗} ⩽c(x,t)⩽c^{∗}$ , $0⩽d_{∗} ⩽d(x,t)⩽d^{∗}$ and $a ̄_{∗} ⩽ ∂ a(x,t) ∂ t ⩽ a ̄^{∗}$ , $b ̄_{∗} ⩽ ∂ b(x,t) ∂ t ⩽ b ̄^{∗}$ , $c ̄_{∗} ⩽ ∂ c(x,t) ∂ t ⩽ c ̄^{∗}$ , $d ̄_{∗} ⩽ ∂ d(x,t) ∂ t ⩽ d ̄^{∗}$ in problem , , , .

Now, let us establish the mesh ω_hτ=ω_h×ω_τ in domain D, such that $ω_{h} ={x_{i} =ih, i=1,2,…,N−1, h=ℓ/N},$ $ω_{τ} ={t_{j} =jτ, j=1,2,…,M−1, τ=T/M}$ and $ω ̄_{h} =ω_{h} ∪{x=0,ℓ}$ , ω_τ⁺=ω_τ∪{t=T}.

We establish the scheme in two stage. Firstly, if the basis functions $ϕ_{i} (x)= ϕ_{i}^{(1)} (x)≡(x−x_{i−1})/h, x_{i−1} <x<x_{i}, ϕ_{i}^{(2)} (x)≡(x_{i+1} −x)/h, x_{i} <x<x_{i+1}, 0, x∉(x_{i−1},x_{i+1}), i=1,2,…,N−1$ is

Error estimates of approximate solution

The following difference problem for the error can be written, while z=y−u, $ℓz=R, (x,t)∈ω_{h} ×ω_{τ}^{+},$ $z(0,t)=z(ℓ,t)=0, t∈ ω ̄_{τ},$ $z(x,0)=z(x,T), x∈ ω ̄_{h},$ $z(x,τ)=z(x,T+τ), x∈ ω ̄_{h} .$

Lemma 3.1

When the condition $λ_{0} c_{∗} + ℓ^{2} 8 d_{∗} − 14 ℵ_{0} − ℓ^{2} 32 ℵ_{1} − 12 c ̄^{∗} + ℓ^{2} 8 d ̄^{∗} >0,$ held, the following estimation for error of difference problem $∥z_{t} ∥^{2} +∥ z ̂_{x̄} ∥^{2} +∥z∥^{2} ⩽Cτ∑_{i=1}^{M} (∥R_{i}^{(0)} ∥^{2} +∥R_{i}^{(1)} ∥^{2}) exp (−C_{1} t_{M−i}), t∈ω_{τ}^{+}$ is true, such that $λ_{0} < 4 ℓ^{2} α+ b_{∗} 2 and ℵ_{0} = max (a ̄^{∗},3 a ̄^{∗}), ℵ_{1} = max (b ̄^{∗},3 b ̄^{∗}),$ where C and C₁, which are independent of h and τ, are positive constants.

Proof

We begin to prove by

Numerical example

The theoretical results are controlled for the following example: $f(x,t)=π^{2} sin πx e^{sin2πt} (4 cos^{2} 2πt−4 sin 2πt+2π cos 2πt+1),$ $a(x,t)=1, b(x,t)=0, c(x,t)=1, d(x,t)=0, T=1, ℓ=1.$ The exact solution of appropriate problem is given by $u(x,t)= sin πx e^{sin2πt} .$ The following iteration process is applied for (2.4) $y_{t̄t,i}^{(n+1)} − Ay_{0tx̄}^{(n+1)}_{x} +By_{0t}^{(n+1)} − Cy_{x̄}^{(n+1)}_{x} +Dy^{(n+1)} =f,$ $y^{(n+1)} (0,t)=y^{(n+1)} (ℓ,t)=0,$ $y^{(n+1)} (x,0)=y^{(n)} (x,T),$ $y^{(n+1)} (x,τ)=y^{(n)} (x,T+τ),$ n=0,1,2,…, y⁽⁰⁾(x,T) and y⁽⁰⁾(x,T+τ) are arbitrary functions.

The computational results

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On difference schemes for problems of the theory of dispersive waves
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G.M. Amiraliyev
Towards the numerical solution of the periodical on time problem for pseudo-parabolic equation
Numer. Meth. Anal. BAKU State Univ
(1988)
G.M. Amiraliyev
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Applied Mathematics and Computation

Numerical solution of linear wave equation with strong dissipative term

Abstract

Introduction

Section snippets

Establish of difference scheme

Error estimates of approximate solution

Numerical example

On difference schemes for problems of the theory of dispersive waves

Soviet Math. Dokl

Towards the numerical solution of the periodical on time problem for pseudo-parabolic equation

Numer. Meth. Anal. BAKU State Univ

Investigation of the difference schemes for the quasi-linear Sobolev equations

Diff. Equat

Experiments of Solitons in Plasmas, Soliton in Action