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it has been shown that classical finite difference schemes fail to give acceptable answers when a is large (typically values between 1000 and 10000). We get so-called spurious oscillations, and this problem is also encountered when solving one-factor and multifactor Black–Scholes equations using finite difference methods. We have resolved this problem using so-called exponentially fitted schemes. We motivate the scheme in the present context, and later chapters describe how to apply it to more complicated cases.

      (2.25)sigma equals StartFraction italic a k left-parenthesis theta plus left-parenthesis 1 minus theta right-parenthesis e Superscript negative italic a k Baseline right-parenthesis Over 1 minus e Superscript negative italic a k Baseline EndFraction period

      Having found the fitting factor for the constant coefficient case, we generalise to a scheme for the case (2.1) as follows:

      (2.26)StartLayout 1st Row 1st Column Blank 2nd Column sigma Superscript n comma theta Baseline StartFraction u Superscript n plus 1 Baseline minus u Superscript n Baseline Over k EndFraction plus a Superscript n comma theta Baseline u Superscript n comma theta Baseline equals f Superscript n comma theta Baseline comma n equals 0 comma ellipsis comma upper N minus 1 comma 0 less-than-or-equal-to theta less-than-or-equal-to 1 2nd Row 1st Column Blank 2nd Column u Superscript 0 Baseline equals upper A 3rd Row 1st Column Blank 2nd Column sigma Superscript n comma theta Baseline equals StartFraction a Superscript n comma theta Baseline left-parenthesis theta plus left-parenthesis 1 minus theta right-parenthesis e Superscript minus a Super Superscript n comma theta Superscript k Baseline right-parenthesis Over 1 minus e Superscript minus a Super Superscript n Superscript comma theta Super Subscript k Superscript Baseline EndFraction k period EndLayout

      In practice we work with a number of special cases:

      (2.27)StartLayout 1st Row 1st Column theta equals 0 2nd Column left-parenthesis implicit right-parenthesis 2nd Row 1st Column sigma Superscript n comma o Baseline equals a Superscript n plus 1 Baseline k slash left-parenthesis e Superscript a Super Superscript n plus 1 Superscript k Baseline minus 1 right-parenthesis 2nd Column Blank 3rd Row 1st Column theta equals one half 2nd Column left-parenthesis fitted upper B o x right-parenthesis 4th Row 1st Column theta Superscript n comma one half Baseline equals StartFraction a Superscript n comma one half Baseline k Over 2 EndFraction left-parenthesis StartFraction 1 plus e Superscript minus a Super Superscript n comma one half Superscript k Baseline Over 1 minus e Superscript minus a Super Superscript n comma one half Superscript k Baseline EndFraction right-parenthesis 2nd Column Blank 5th Row 1st Column equals StartFraction a Superscript n comma one half Baseline k Over 2 EndFraction italic hyperbolic cotangent StartFraction a Superscript n comma one half Baseline k Over 2 EndFraction 2nd Column left-parenthesis Il prime in right-parenthesis period EndLayout right-brace

      In the final case italic hyperbolic cotangent left-parenthesis x right-parenthesis is the hyperbolic cotangent function.

      2.4.2 Scalar Non-Linear Problems and Predictor-Corrector Method

      Real-life problems are very seldom linear. In general, we model applications using non-linear IVPs:

      (2.29) Скачать книгу