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initial value problem and is chosen in such a way that it produces an exact solution for a certain model problem. To this end, let us examine the scalar ODE:

      (2.41)StartFraction italic d y Over italic d t EndFraction equals f left-parenthesis t comma y left-parenthesis t right-parenthesis right-parenthesis comma t element-of left-parenthesis 0 comma upper T right-bracket

      and let us approximate it using the Theta method:

      where the parameter theta has not yet been specified. We determine it using the heuristic that this so-called Theta method should be exact for the linear constant-coefficient model problem:

      (2.44)StartLayout 1st Row 1st Column Blank 2nd Column y Subscript n plus 1 Baseline equals StartFraction 1 plus normal upper Delta t normal lamda Over 1 minus left-parenthesis 1 minus theta right-parenthesis t normal lamda EndFraction y Subscript n Baseline 2nd Row 1st Column Blank 2nd Column and 3rd Row 1st Column Blank 2nd Column theta equals minus StartFraction 1 Over normal upper Delta t normal lamda EndFraction minus StartFraction exp left-parenthesis normal upper Delta t normal lamda right-parenthesis Over 1 minus exp left-parenthesis normal upper Delta t lamda right-parenthesis EndFraction period EndLayout

      We need to determine if this scheme is stable (in some sense). To answer this question, we introduce some concepts.

      Definition 2.3 The region of (absolute) stability of a numerical method for an initial value problem is the set of complex values normal upper Delta t normal lamda for which all discrete solutions of the model problem (2.43) remain bounded when n approaches infinity.

      Definition 2.4 A numerical method is said to be A-stable if its region of stability is the left-half plane, that is:

upper R equals left-brace normal upper Delta t normal lamda element-of normal double struck upper C semicolon italic upper R e normal upper Delta t normal lamda less-than 0 right-brace period

      Returning to the exponentially fitted method, we can check that it is A-stable because for all normal upper Delta t normal lamda less-than 0 we have theta less-than-or-equal-to one half, and this condition can be checked using the scheme (2.42) for the model problem (2.43).

      We can generalise the exponential fitting technique to linear and non-linear systems of equations. In the case of a linear system, stiffness is caused by an isolated real negative eigenvalue of the matrix A in the equation:

      where y comma f element-of normal double struck upper R Superscript n and A is a constant n times n matrix with eigenvalues normal lamda Subscript j Baseline element-of normal double struck upper C and eigenvectors x Subscript j Baseline element-of normal double struck upper C Superscript n Baseline comma j equals 1 comma ellipsis comma n period

y left-parenthesis t right-parenthesis equals sigma-summation Underscript j equals 1 Overscript n Endscripts c Subscript j Baseline exp left-parenthesis normal lamda Subscript j Baseline t right-parenthesis x Subscript j Baseline plus g left-parenthesis t right-parenthesis

      where c Subscript j Baseline comma j equals 1 comma ellipsis comma n are arbitrary constants and g left-parenthesis t right-parenthesis is a particular integral. In this case we can employ exponential fitting by fitting the dominant eigenvalues which can be computed by the Power method, for example.

      If we assume that the real parts of the eigenvalues are less than zero, we can conclude that the solution tends to the steady-state. Even though this solution is well-behaved the cause of numerical instabilities is the presence of quickly decaying transient components of the solution caused by the dominant eigenvalues of the matrix A in (2.45).

StartLayout 1st Row 1st Column Blank 2nd Column StartFraction d Over italic d t EndFraction StartBinomialOrMatrix y 1 Choose y 2 EndBinomialOrMatrix equals Start 2 By 2 Matrix 1st Row 1st Column minus alpha 1 2nd Column 0 2nd Row 1st Column 0 2nd Column minus alpha 2 EndMatrix StartBinomialOrMatrix y 1 Choose y 2 EndBinomialOrMatrix 2nd Row 1st Column Blank 2nd Column y 1 left-parenthesis 0 right-parenthesis equals y 2 left-parenthesis 0 right-parenthesis equals 1 comma 0 less-than-or-equal-to alpha 2 less-than less-than alpha 1 period EndLayout

      The solution of this system is given by:

StartLayout 1st Row 1st Column y 1 left-parenthesis t right-parenthesis 2nd Column equals e Superscript minus alpha 1 t Baseline 2nd Row 1st Column y 2 left-parenthesis t 
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