ΩΩΩΩΩ

The initial condition and the Dirichlet boundary conditions are defined by means of the exact solution (6.94) at starting time T0 = 100 and with u0 = 1.0. We have λ = 10−5, v = 0.001, and D = 0.0001.

Introduction and Solution using classical methods

In order to solve the model problem using the overlapping Schwarz waveform relaxation method, we divide the domain Ω into two overlapping subdomains Ω1 = [0, L2] and Ω2 = [L1, L], where L1 < L2, and Ω1

⋂ Ω2 = [L1, L2]

is the overlapping region for Ω1 and Ω2. To start the waveform relaxation algorithm we consider first the solution

vt = Dvxx − vvx − λv over Ω1 , t ∈ [T0, Tf ], v(0, t) = f1(t) , t ∈ [T0, Tf ], v(L2, t) = w(L2, t) , t ∈ [T0, Tf ], v(x, T0) = u0 x ∈ Ω1,

(6.95)

and

wt = Dwxx − vwx − λw over Ω2 , t ∈ [T0, Tf ], w(L1, t) = v(L1, t) , t ∈ [T0, Tf ], w(L, t) = f2(t) , t ∈ [T0, Tf ], w(x, T0) = u0 x ∈ Ω2,

(6.96)

where v(x, t) = u(x, t)|Ω1 and w(x, t) = u(x, t)|Ω2 . Then the Schwarz waveform relaxation is given by

vk+1t = Dvk+1xx − vvk+1x − λvk+1 over Ω1 , t ∈ [T0, Tf ], vk+1(0, t) = f1(t) , t ∈ [T0, Tf ], vk+1(L2, t) = wk(L2, t) , t ∈ [T0, Tf ], vk+1(x, T0) = u0 x ∈ Ω1,

(6.97)

and

wk+1t = Dw k+1 xx − vwk+1x − λwk+1 over Ω2 , t ∈ [T0, Tf ],

wk+1(L1, t) = vk(L1, t) , t ∈ [T0, Tf ], wk+1(L, t) = f2(t) , t ∈ [T0, Tf ], wk+1(x, T0) = u0 x ∈ Ω2.