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<a name="Linearization-and-discretization"></a>
<ul class="toc"><li><a href="mini_ker.html#Top">Miniker 102 manual</a> </li>
<li><ul class="toc"><li><a href="TEF-overview.html#TEF-overview">1. An overview of the <acronym>TEF</acronym> formalism</a> </li>
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<a name="Linearization-and-discretization-in-the-TEF"></a>
<h2 class="section"> 1.2 Linearization and discretization in the <acronym>TEF</acronym> </h2>
<p>The relations between sub-systems is excessively difficult to exhibit when
having to cope with non-linear system. In the <acronym title="Transfer Evolution Formalism">TEF</acronym>, the
<acronym title="Tangent Linear System">TLS</acronym> (Tangent Linear System) is constructed along the trajectory.
One considers the system over a small portion along the trajectory, say
between t and t + δt. The variation δη
of η and δφ of φ is obtained
through a Padé approximation of the state-transition matrix. The final
form of the algebraic system is closed to the classical Crank-Nicolson scheme:
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<center class="math-display" >
<img
src="mini_ker_tex4ht_tex2x.png" alt="( )( ) ( )
A B δη Γ
- C+ I - D δφ = Ω " class="math-display" ></center>
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<p>The blocks appearing in the Jacobian matrix are constructed with partial derivative
of f and g, and with δt. From this system the
elimination of δη leads to another formulation giving
the coupling between transfers, and allows for the δφ
computation. The δφ value is then substitued in
δη to complete the time-step solving process.
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