multiple time scale dynamics with two fast variables and one slow ...

This explicit solution provides a benchmark for evaluating the accuracy of the
algorithm described above. The slow manifold of the system is the line{y=
x1+ǫ, x2= 0} containing the trajectories (x1, x2, y)(t)=(y(0)−ǫ+ t, 0, y(0)+t).
The discretized equations of the algorithm can also be solved explicitly for
system (4.3). The first step in doing so is to observe that the equations for x1 and
y are separable from those for x2, and this remains the case for the discretized
equations of the boundary value solver. Substituting the equations for the y-
variable into the boundary value equations produces the equation y j+1− y j =
t j+1− t j on each mesh interval. If a boundary condition is imposed on one end
of the time interval [a, b], these equations yield a solution that is a discretization
of an exact solution of the differential equation. Convergence occurs in a single
step.
Assume now that y j+1− y j = t j+1− t j and set w j = y j− (x1) j−ǫ to be the
difference between the x1 coordinate of a point and a point of the slow manifold.
The boundary value equations become
δ 2 − 6δǫ+ 12ǫ 2
8δǫ 2
w j− δ2 + 6δǫ+ 12ǫ 2
8δǫ 2 w j+1= 0 (4.4)
for a uniform mesh withδ=y j+1−yj= t j+1−tj. The boundary conditions at t0= a
must be transverse to the x1 coordinate axis which is E s . Therefore, we choose to
fix the value of x1 as the boundary condition at t0= a. Note that these equations
are satisfied when the w j vanish, so if the value of x1 is y−ǫ at t0=a, the w j
yield a discretization of the exact solution along the slow manifold. Solving the
equation (4.4) for w j+1 in terms of w j yields
w j+1= δ2 − 6δǫ+ 12ǫ 2
δ 2 + 6δǫ+ 12ǫ 2 w j
Like the solutions of the differential equation, the values w j decrease exponen-
tially as a function of time. The ratioρ j=w j+1/w j is a function of (δ/ǫ) whose
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