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only if StartAbsoluteValue 1 plus alpha EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue 1 minus alpha EndAbsoluteValue and this implies italic upper R e left-parenthesis StartFraction mu normal upper Delta t Over 2 EndFraction right-parenthesis less-than-or-equal-to 0 (recall that mu is a complex number) where italic upper R e left-parenthesis z right-parenthesis means ‘real part of z’.

      We now discuss convergence issues. We say that a difference scheme has order of accuracy p if:

max StartAbsoluteValue upper X Subscript n Baseline minus upper X left-parenthesis t Subscript n Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to upper M normal upper Delta t Superscript p Baseline comma for 0 less-than-or-equal-to n less-than-or-equal-to upper N

      where upper X Subscript n Baseline equals approximate solution of (2.32), upper X left-parenthesis t Subscript n Baseline right-parenthesis equals exact solution of (2.31), and upper M is independent of normal upper Delta t.

      We conclude this section by stating a convergence result that allows us to estimate the error between the exact solution of an initial value problem and the solution of a multistep scheme that approximates it. To this end, we consider the n-dimensional autonomous initial value problem:

left-parenthesis IVPI right-parenthesis StartLayout Enlarged left-brace 1st Row y prime equals StartFraction italic d y Over italic d t EndFraction equals f left-parenthesis y right-parenthesis comma in the interval left-bracket a comma b right-bracket comma y left-parenthesis a right-parenthesis equals c 2nd Row where colon 3rd Row y equals left-parenthesis y 1 comma ellipsis comma y Subscript n Baseline right-parenthesis Superscript down-tack Baseline 4th Row f left-parenthesis y right-parenthesis equals left-parenthesis f 1 left-parenthesis y right-parenthesis comma ellipsis comma f Subscript n Baseline left-parenthesis y right-parenthesis right-parenthesis Superscript down-tack Baseline comma c equals left-parenthesis c 1 comma ellipsis comma c Subscript n Baseline right-parenthesis Superscript down-tack Baseline period EndLayout

      By autonomous we mean that f left-parenthesis y right-parenthesis is a function of the dependent variable y only and is thus not of the form f left-parenthesis y comma t right-parenthesis. The latter form is called non-autonomous.

      We approximate this IVP using the multistep method (2.32). We recall:

sigma-summation Underscript j equals 0 Overscript k Endscripts left-parenthesis alpha Subscript j Baseline upper X Subscript n minus j Baseline minus normal upper Delta t beta Subscript j Baseline f left-parenthesis upper X Subscript n minus j Baseline right-parenthesis right-parenthesis equals 0 period

      Theorem 2.2 Assume that the solution normal y of the italic IVPI is p plus 1 times differentiable with double-vertical-bar y Superscript left-parenthesis p plus 1 right-parenthesis Baseline left-parenthesis x right-parenthesis double-vertical-bar less-than-or-equal-to upper K 0 comma p greater-than-or-equal-to 1 and assume that f is differentiable for all y.

      Suppose furthermore that the sequence left-brace upper X Subscript n Baseline right-brace is defined by the equations:

StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column upper X Subscript n Baseline equals y left-parenthesis t Subscript n Baseline right-parenthesis plus normal epsilon Subscript n Baseline comma n equals 0 comma ellipsis comma normal k minus 1 2nd Row 1st Column Blank 2nd Column Blank 3rd Column sigma-summation Underscript j equals 0 Overscript k Endscripts left-parenthesis alpha Subscript j Baseline upper X Subscript n minus j Baseline minus normal upper Delta Subscript t Baseline beta Subscript j Baseline f left-parenthesis upper X Subscript n minus j Baseline right-parenthesis right-parenthesis equals normal epsilon Subscript n comma Baseline k less-than-or-equal-to n less-than-or-equal-to StartFraction b minus a Over normal upper Delta t EndFraction period EndLayout

      If the multistep method is stable and satisfies:

sigma-summation Underscript j equals 1 Overscript k Endscripts left-parenthesis alpha Subscript j Baseline y left-parenthesis t minus j normal upper Delta t right-parenthesis minus normal upper Delta t beta Subscript j Baseline y prime left-parenthesis t minus italic j h right-parenthesis tilde upper C normal upper Delta t Superscript p plus 1 Baseline y Superscript p plus 1 Baseline left-parenthesis t right-parenthesis for-all t element-of left-bracket 0 comma upper T right-bracket

      where C is a positive constant independent of Скачать книгу