Numerical Methods in Computational Finance - Daniel J. Duffy

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greater-than-or-equal-to 0 comma n equals 0 comma ellipsis comma upper N minus 1 comma w Superscript 0 Baseline greater-than-or-equal-to 0"/>

      implies that w Superscript n Baseline greater-than-or-equal-to 0 for-all n equals 0 comma ellipsis comma upper N. Here, w Superscript n is a mesh function defined at the mesh points t Subscript n.

      Based on this definition, we see that the implicit Euler scheme is always positive while the explicit Euler scheme is positive if the term:

      Definition 2.2 A difference scheme is stable if its solution is based in much the same way as the solution of the continuous problem (2.1) (see Theorem 2.1), that is:

      (2.19)StartAbsoluteValue u Superscript n Baseline EndAbsoluteValue less-than-or-equal-to StartFraction upper N Over alpha EndFraction plus StartAbsoluteValue upper A EndAbsoluteValue comma n greater-than-or-equal-to 0

      where:

a left-parenthesis t Subscript n Baseline right-parenthesis greater-than-or-equal-to alpha comma n greater-than-or-equal-to 0 comma StartAbsoluteValue f left-parenthesis t Subscript n Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to upper N comma n greater-than-or-equal-to 0

      and:

u Superscript 0 Baseline equals upper A period

      (2.20)StartAbsoluteValue u Superscript n Baseline minus u left-parenthesis t Subscript n Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to italic upper M k Superscript p Baseline comma p equals 1 comma 2 comma ellipsis comma n greater-than-or-equal-to 0

      (2.21)StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column Implicit Euler colon StartAbsoluteValue u Superscript n Baseline minus u left-parenthesis t Subscript n Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to italic upper M k comma n equals 0 comma ellipsis comma upper N 2nd Row 1st Column Blank 2nd Column Blank 3rd Column Crank en-dash Nicolson left-parenthesis Box right-parenthesis colon StartAbsoluteValue u Superscript n Baseline minus u left-parenthesis t Subscript n Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to italic upper M k squared comma n equals 0 comma ellipsis comma upper N 3rd Row 1st Column Blank 2nd Column Blank 3rd Column Explicit Euler colon StartAbsoluteValue u Superscript n Baseline minus u left-parenthesis t Subscript n Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to italic upper M k comma n equals 0 comma ellipsis comma upper N if 1 minus a Superscript n Baseline k greater-than 0 period EndLayout

      Thus, we see that the Box method is second-order accurate and is better than the implicit Euler scheme, which is only first-order accurate.

      We introduce exponentially fitted schemes that are used for boundary layer problems (for example, convection-dominated PDEs) and the extrapolation method to increase the accuracy of finite difference schemes. We shall see how to apply these techniques to more complex problems in later chapters. We also discuss predictor-corrector methods.

      2.4.1 Exponential Fitting

      We now introduce a special class of schemes with desirable properties. These are schemes that are suitable for problems with rapidly increasing or decreasing solutions. In the literature these are called stiff or singular perturbation problems (see Duffy (1980)). We can motivate these schemes in the present context. Let us take the problem (2.1) when a left-parenthesis t right-parenthesis is constant and f left-parenthesis t right-parenthesis is zero. The solution u left-parenthesis t right-parenthesis is given by a special case of (2.2), namely:

      If a is large then the derivatives of u left-parenthesis t right-parenthesis tend to increase; in fact, at t equals 0, the derivatives are given by:

      (2.23)StartFraction d Superscript k Baseline u left-parenthesis 0 right-parenthesis Over italic d t Superscript k Baseline EndFraction equals upper A left-parenthesis negative a right-parenthesis Superscript k Baseline comma k equals 0 comma 1 comma 2 comma ellipsis

      The physical interpretation of this fact is that a boundary layer exits near t equals 0 where u is changing rapidly,

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