Single Shooting

Single shooting is based on the principle of superposition valid for linear ODEs,^e i.e., any linear combination v(z) = $\alpha_{1}^{}$v₁(z) + $\alpha_{2}^{}$v₂(z) of two individual solutions v₁(z) and v₂(z) satisfying the ODE v^$\prime$(z) = $\underline{\mathbf{H}}$(z)v(z) is again a solution of it. The basic theory of ODEs states that an N-dimensional ODE system has exactly N linearly independent solution vectors v_i(z) with i = 1,..., N [200, pp. 85-93]. The BCs can be chosen arbitrarily provided that the vectors v_i(z) are linearly independent and thus span the whole N-dimensional vector space $\mathbb{C}$^N. By taking the unit vectors e_i, i.e., [e_i]_j = $\delta_{ij}^{}$, as initial values at one boundary point, e.g., at z = 0, the obtained solutions v_i(z) are called fundamental solutions,

$$\mathbf{v}_i^\prime(z) = \underline{\mathbf{H}}(z)\mathbf{v}_i(z)\quad\qquad\text{with}\qquad\quad\mathbf{v}_i(0) = \mathbf{e}_i,$$ (6.36)

that are arranged in the N x N-dimensional fundamental solution matrix V(z). The solution u(z) of any first-order, N-dimensional BVP can be represented by a specific superposition of the fundamental solutions, i.e.,

 

$$\mathbf{u}(z) = \sum_{i=1}^N s_i \mathbf{v}_i(z)\quad\qquad\text{or}\qquad\quad\mathbf{u}(z) = \underline{\mathbf{V}}(z)\mathbf{s},$$ (6.37)

with

$$\underline{\mathbf{V}}(z) = (\mathbf{v}_1(z) \; \mathbf{v}_2(z) \... ...quad\text{and}\quad\qquad\mathbf{s} = (s_1 \; s_2 \;\cdots\; s_N)^{\mathrm{T}}.$$ (6.38)

The coefficients s_i have to be chosen so that the imposed BCs are satisfied, i.e., in our case they take the form (cf. (6.51))

$$\underline{\mathbf{B}}_0\underline{\mathbf{V}}(0)\mathbf{s}^{pq} ... ...d\underline{\mathbf{B}}_h\underline{\mathbf{V}}(h)\mathbf{s}^{pq} = \mathbf{0}.$$ (6.39)

These two equations build up a linear algebraic system. Due to the multiple excitation vectors a^pq we obtain an algebraic equation with N_s right-hand sides and thus N_s coefficient vectors s^pq, i.e.,

 

$$\begin{bmatrix}\underline{\mathbf{B}}_0\\ \underline{\mathbf{B}}_... ... = \begin{bmatrix}\underline{\mathbf{A}}\\ \underline{\mathbf{0}}\end{bmatrix},$$ (6.40)

whereby all the N_s vectors a^pq and s^pq are grouped together in the two matrices $\underline{\mathbf{A}}$ and $\underline{\mathbf{S}}$, respectively (cf. (6.54)). Note that $\underline{\mathbf{V}}$(0) = $\underline{\mathbf{I}}$ has already been inserted for the fundamental solutions at z = 0. (6.59) can be efficiently solved with LU-factorization [202, pp. 94-104] to obtain simultaneously all the coefficients s^pq looked for. With them the solution $\underline{\mathbf{U}}$(z) can directly be calculated from (6.56) if the complete fundamental solution matrix $\underline{\mathbf{V}}$(z) is stored for all interesting vertical positions. Otherwise the initial values at z = 0 are simply given by $\underline{\mathbf{U}}$(0) = $\underline{\mathbf{S}}$ since $\underline{\mathbf{V}}$(0) = $\underline{\mathbf{I}}$ and the IVP thus obtained can easily be integrated. An algorithm implementing the above procedure is described in Table 6.2. As already mentioned this algorithm suffers from potential instabilities [200, pp. 137-143] and has thus to be further refined for our problem.

  

Table 6.2: Algorithm for single shooting.

Footnotes

... ODEs,^e

This does not mean that shooting is restricted to linear ODEs. In the linear case, however, the theory can be presented in a very compact and elegant way, whereas in the nonlinear case the treatment is more complicated and additional steps are required. Since our problem is linear we restrict the discussion to this simpler situation.