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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 140415, 5260]*) (*NotebookOutlinePosition[ 141111, 5284]*) (* CellTagsIndexPosition[ 141067, 5280]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Runge-Kutta methods \ \>", "Title"], Cell["Explicit methods", "Subtitle"], Cell[CellGroupData[{ Cell["\<\ First, evaluate the following cell to load all necessary functions. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(<< \ NumericalMath`Butcher`\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(FinalStep[s_]\ := \ x0\ + \ h\ Sum[b[i]\ f[k[i]], \ {i, \ s}]\)\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(ExplicitMethod[ s_]\ := \ \[IndentingNewLine]Append[\[IndentingNewLine]Table[ k[i]\ == \ x0 + \ h\ Sum[a[i, j]\ f[k[j]], \ {j, \ i - 1}], \ {i, s}], \[IndentingNewLine]x1\ \[Equal] \ FinalStep[s]]\)], "Input",\ InitializationCell->True], Cell[BoxData[ \(DiagonalImplicitMethod[ s_]\ := \ \[IndentingNewLine]Append[\[IndentingNewLine]Table[ k[i]\ == \ x0\ + \ h\ Sum[a[i, j]\ f[k[j]], \ {j, \ i}], \ {i, s}], \[IndentingNewLine]x1\ \[Equal] \ FinalStep[s]]\)], "Input",\ InitializationCell->True], Cell[BoxData[ \(ImplicitMethod[ s_]\ := \ \[IndentingNewLine]Append[\[IndentingNewLine]Table[ k[i]\ == \ x0\ + \ h\ Sum[a[i, j]\ f[k[j]], \ {j, \ s}], \ {i, s}], \[IndentingNewLine]x1\ \[Equal] \ FinalStep[s]]\)], "Input",\ InitializationCell->True], Cell[BoxData[ \(MethodAsArray[meth_]\ := \ Map[\((Part[#, 1] - Part[#, 2])\) &, meth]\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(MethodVariables[meth_]\ := \ Map[First, \ meth]\)\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(DoStep[meth_, \ mvs_, \ x_]\ := \ x1\ /. \[IndentingNewLine]With[{args\ = \ Prepend[Map[{#, x} &, mvs], \ meth\ /. \ x0 \[Rule] x]}, \[IndentingNewLine]Apply[ FindRoot, \ args]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(SolveWithMethod[meth_, \ x_, \ steps_]\ := \[IndentingNewLine]With[{ar\ = \ MethodAsArray[meth], \[IndentingNewLine]mvs\ = \ MethodVariables[ meth]}, \[IndentingNewLine]NestList[\((DoStep[ar, \ mvs, \ #])\) &, \ x, \ steps]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(SubstituteRowSumsExplicit[oc_, \ s_]\ := \ oc\ /. \ {c[i_]\ \[RuleDelayed] \ \ Sum[ a[i, j], \ {j, \ i - 1}]}\)\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(SubstituteRowSumsImplicit[oc_, \ s_]\ := \ oc\ /. \ {c[i_]\ \[RuleDelayed] \ \ Sum[ a[i, j], \ {j, \ s}]}\)\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(SubstituteRowSumsDiagonalImplicit[oc_, \ s_]\ := \ oc\ /. \ {c[i_]\ \[RuleDelayed] \ \ Sum[ a[i, j], \ {j, \ i}]}\)\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(ExtendCoefficients[coef_, ex_]\ := \ Join[coef\ /. \ ex, \ ex]\)\)\)], "Input", InitializationCell->True] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We will compute the order conditions of a two stage, second order explicit \ Runge-Kutta method. The first argument given to the RungeKuttaOrderConditions function indicates \ the desired order, the second argument indicates the desired number of stages.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(orderconditions\ = \ Map[First, RungeKuttaOrderConditions[2, 2, \ RungeKuttaMethod\ \[Rule] \ Explicit]]\)], "Input"], Cell[BoxData[ \({b\_1 + b\_2 == 1, b\_2\ c\_2 == 1\/2}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We replace the row sums c[i] by their explicit values in terms of a[i].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(orderconditions2\ = \ SubstituteRowSumsExplicit[orderconditions, \ 2]\)\)\)], "Input"], Cell[BoxData[ \({b\_1 + b\_2 == 1, a\_\(2, 1\)\ b\_2 == 1\/2}\)], "Output"] }, Closed]] }, Open ]], Cell["We compute the principal errors.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(ButcherPrincipalError[2, 2, RungeKuttaMethod \[Rule] Explicit]\)\)\)], "Input"], Cell[BoxData[ \({\(-\(1\/6\)\), 1\/2\ \((\(-\(1\/3\)\) + b\_2\ c\_2\%2)\)}\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ There is effectively one degree of freedom. Therefore, we add the condition \ b[2] c[2]^2=1/3 to minimalise the principal error.].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(orderconditions3\ = \ Append[orderconditions2, \ b[2] a[2, 1]^2 \[Equal] 1/3]\)\)\)], "Input"], Cell[BoxData[ \({b\_1 + b\_2 == 1, a\_\(2, 1\)\ b\_2 == 1\/2, a\_\(2, 1\)\%2\ b\_2 == 1\/3}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We try to find a solution.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(coefficients\)\(\ \)\(=\)\(\ \)\(Solve[ orderconditions3, \ {b[1], \ b[2], \ a[2, 1]}]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \({{b\_1 \[Rule] 1\/4, b\_2 \[Rule] 3\/4, a\_\(2, 1\) \[Rule] 2\/3}}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The generic 2-stage explicit Runge-Kutta method has the following form.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(meth\ = \ ExplicitMethod[2]\)], "Input"], Cell[BoxData[ \({k[1] == x0, k[2] == x0 + h\ a\_\(2, 1\)\ f[k[1]], x1 == x0 + h\ \((b\_1\ f[k[1]] + b\_2\ f[k[2]])\)}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["We substitute the coefficients into this method.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(meth2\ = \ meth\ /. \ First[coefficients]\)], "Input"], Cell[BoxData[ \({k[1] == x0, k[2] == x0 + 2\/3\ h\ f[k[1]], x1 == x0 + h\ \((1\/4\ f[k[1]] + 3\/4\ f[k[2]])\)}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We also substitute a concrete choice for the function f and stepsize h.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(meth3\ = \ meth2\ /. \ {\ f[x_]\ -> \ 1\ + x^2 - x^3, \ h\ -> \ 0.01}\)], "Input"], Cell[BoxData[ \({k[1] == x0, k[2] == x0 + 0.006666666666666667`\ \((1 + k[1]\^2 - k[1]\^3)\), x1 == x0 + 0.01`\ \((1\/4\ \((1 + k[1]\^2 - k[1]\^3)\) + 3\/4\ \((1 + k[2]\^2 - k[2]\^3)\))\)}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We will now perform 100 time steps with this method. I.e.we integrate from \ x=0 to x=1. Note that this computation takes a while; obviously, it would be \ faster if this program was written in Fortran rather than in Mathematica. ;-)\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(solution\ = \ SolveWithMethod[meth3, \ 0, 500]\)], "Input"], Cell[BoxData[ \({0, 0.01`, 0.020001975097515`, 0.030007836810249558`, 0.04001943951749581`, 0.05003858163344705`, 0.06006700662201199`, 0.07010640385916989`, 0.08015840934844069`, 0.0902246062946282`, 0.10030652554062543`, 0.11040564587174723`, 0.12052339419177313`, 0.1306611455746423`, 0.1408202231955396`, 0.1510018981449464`, 0.1612073891291012`, 0.17143786206022035`, 0.1816944295397701`, 0.19197815023805467`, 0.20229002817339164`, 0.21263101189418568`, 0.22300199356728148`, 0.2334038079760805`, 0.24383723143203873`, 0.2543029806033268`, 0.2648017112646288`, 0.27533401697227955`, 0.2859004276691927`, 0.29650140822431453`, 0.30713735691164645`, 0.3178086038342149`, 0.3285154092987284`, 0.33925796214704684`, 0.35003637805099436`, 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h@0000<00^4200?Q1004h@030>7Q03OQ00<0hN40c^4000?Q00<0hN401n400`3Qh@05h@030>7Q03SQ 00<0hN40cN4000?Q00<0hN402>400`3Qh@04h@<0>N400`3Qh@37Q00?Q 00<0hN40>N400`3Qh@37Q00CQ00<0hN40>^400`3Qh@3;h@000^4200[Q 0P05h@030>7Q03_Q00<0hN40b^4001GQ00<0hN40>n420<_Q000Eh@80?^400`3Qh@38h@005N400`3Q h@0mh@80bN4001GQ00<0hN40?^400`3Qh@37h@005N400`3Qh@0oh@030>7Q07Q043Q 00<0hN40aN4001GQ00<0hN40@N400`3Qh@34h@005N4204;Q0P35h@005N400`3Qh@12h@80a>4001GQ 00<0hN40@n4207Q04CQ0P32h@005N400`3Qh@16h@030>7Q0;oQ000Eh@80B>400`3Q h@2nh@005N400`3Qh@18h@80_^4001GQ00<0hN40BN420;gQ000Eh@030>7Q04_Q0P2kh@0000?Q0000 0`02h@801N4300?Q00<0hN40C>430;WQ0003h@030>7Q00[Q00<0hN400^400`3Qh@1>h@<0]n4000?Q 00<0hN401n4500?Q0`1Ah@80]N4000?Q00<0hN401n40103Qh@04h@030>7Q05?Q0`2bh@000n400`3Q h@08h@030>4000CQ00<0hN40E^430:oQ0002h@802n4200CQ00<0hN40FN440:_Q000Eh@030>7Q05gQ 1P2Uh@005N4206CQ2@2Lh@005N400`3Qh@1/hA/0PN4001GQ00<0hN40Q^5i00WQ000Eh@030>7Q0?oQ 2N4001GQ00<0hN40on49h@005N420?oQ2^4001GQ00<0hN40on49h@00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-41.6489, -0.121826, \ 1.94839, 0.00922205}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]] }, Open ]], Cell["\<\ Implicit methods\ \>", "Subtitle"], Cell[CellGroupData[{ Cell["\<\ We will compute the order conditions of a two stage, fourth order implicit Runge-Kutta method. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(orderconditions\ = \ Map[First, RungeKuttaOrderConditions[4, 2, \ RungeKuttaMethod\ \[Rule] \ Implicit]]\)], "Input"], Cell[BoxData[ \({b\_1 + b\_2 == 1, b\_1\ c\_1 + b\_2\ c\_2 == 1\/2, b\_1\ \((a\_\(1, 1\)\ c\_1 + a\_\(1, 2\)\ c\_2)\) + b\_2\ \((a\_\(2, 1\)\ c\_1 + a\_\(2, 2\)\ c\_2)\) == 1\/6, b\_1\ \((a\_\(1, 1\)\ \((a\_\(1, 1\)\ c\_1 + a\_\(1, 2\)\ c\_2)\) + a\_\(1, 2\)\ \((a\_\(2, 1\)\ c\_1 + a\_\(2, 2\)\ c\_2)\))\) + b\_2\ \((a\_\(2, 1\)\ \((a\_\(1, 1\)\ c\_1 + a\_\(1, 2\)\ c\_2)\) + a\_\(2, 2\)\ \((a\_\(2, 1\)\ c\_1 + a\_\(2, 2\)\ c\_2)\))\) == 1\/24}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We replace the row sums c[i] by their explicit values in terms of a[i]. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(orderconditions2\ = \ SubstituteRowSumsImplicit[orderconditions, \ 2]\)\)\)], "Input"], Cell[BoxData[ \({b\_1 + b\_2 == 1, \((a\_\(1, 1\) + a\_\(1, 2\))\)\ b\_1 + \((a\_\(2, 1\) + a\_\(2, 2\))\)\ b\_2 == 1\/2, \((a\_\(1, 1\)\ \((a\_\(1, 1\) + a\_\(1, 2\))\) + a\_\(1, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\)\ b\_1 + \((\((a\_\(1, 1\) + a\_\(1, 2\))\)\ a\_\(2, 1\) + a\_\(2, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\)\ b\_2 == 1\/6, \((a\_\(1, 1\)\ \((a\_\(1, 1\)\ \((a\_\(1, 1\) + a\_\(1, 2\))\) + a\_\(1, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\) + a\_\(1, 2\)\ \((\((a\_\(1, 1\) + a\_\(1, 2\))\)\ a\_\(2, 1\) + a\_\(2, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\))\)\ b\_1 + \((a\_\(2, 1\)\ \((a\ \_\(1, 1\)\ \((a\_\(1, 1\) + a\_\(1, 2\))\) + a\_\(1, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\) + a\_\(2, 2\)\ \((\((a\_\(1, 1\) + a\_\(1, 2\))\)\ a\_\(2, 1\) + a\_\(2, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\))\)\ b\_2 == 1\/24}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Or a bit nicer formatted: \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(orderconditions2\ // \ TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(b\_1 + b\_2 == 1\)}, {\(\((a\_\(1, 1\) + a\_\(1, 2\))\)\ b\_1 + \((a\_\(2, 1\) + a\_\(2, 2\))\)\ b\_2 == 1\/2\)}, {\(\((a\_\(1, 1\)\ \((a\_\(1, 1\) + a\_\(1, 2\))\) + a\_\(1, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\)\ b\_1 + \((\((a\_\(1, 1\) + a\_\(1, 2\))\)\ a\_\(2, 1\) + a\_\(2, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\)\ b\_2 == 1\/6\)}, {\(\((a\_\(1, 1\)\ \((a\_\(1, 1\)\ \((a\_\(1, 1\) + a\_\(1, 2\))\) + a\_\(1, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\) + a\_\(1, 2\)\ \((\((a\_\(1, 1\) + a\_\(1, 2\))\)\ a\_\(2, 1\) + a\_\(2, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\))\)\ b\_1 + \((a\_\(2, 1\)\ \ \((a\_\(1, 1\)\ \((a\_\(1, 1\) + a\_\(1, 2\))\) + a\_\(1, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\) + a\_\(2, 2\)\ \((\((a\_\(1, 1\) + a\_\(1, 2\))\)\ a\_\(2, 1\) + a\_\(2, 2\)\ \((a\_\(2, 1\) + a\_\(2, 2\))\))\))\)\ b\_2 == 1\/24\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { Equal[ Plus[ b[ 1], b[ 2]], 1], Equal[ Plus[ Times[ Plus[ a[ 1, 1], a[ 1, 2]], b[ 1]], Times[ Plus[ a[ 2, 1], a[ 2, 2]], b[ 2]]], Rational[ 1, 2]], Equal[ Plus[ Times[ Plus[ Times[ a[ 1, 1], Plus[ a[ 1, 1], a[ 1, 2]]], Times[ a[ 1, 2], Plus[ a[ 2, 1], a[ 2, 2]]]], b[ 1]], Times[ Plus[ Times[ Plus[ a[ 1, 1], a[ 1, 2]], a[ 2, 1]], Times[ a[ 2, 2], Plus[ a[ 2, 1], a[ 2, 2]]]], b[ 2]]], Rational[ 1, 6]], Equal[ Plus[ Times[ Plus[ Times[ a[ 1, 1], Plus[ Times[ a[ 1, 1], Plus[ a[ 1, 1], a[ 1, 2]]], Times[ a[ 1, 2], Plus[ a[ 2, 1], a[ 2, 2]]]]], Times[ a[ 1, 2], Plus[ Times[ Plus[ a[ 1, 1], a[ 1, 2]], a[ 2, 1]], Times[ a[ 2, 2], Plus[ a[ 2, 1], a[ 2, 2]]]]]], b[ 1]], Times[ Plus[ Times[ a[ 2, 1], Plus[ Times[ a[ 1, 1], Plus[ a[ 1, 1], a[ 1, 2]]], Times[ a[ 1, 2], Plus[ a[ 2, 1], a[ 2, 2]]]]], Times[ a[ 2, 2], Plus[ Times[ Plus[ a[ 1, 1], a[ 1, 2]], a[ 2, 1]], Times[ a[ 2, 2], Plus[ a[ 2, 1], a[ 2, 2]]]]]], b[ 2]]], Rational[ 1, 24]]}]]], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We try to find a solution. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(coefficients\ = \ First[Solve[ orderconditions2, \ {b[1], \ b[2], \ a[1, 1], a[1, 2]}]\ ]\)], "Input"], Cell[BoxData[ \({b\_2 \[Rule] 1 - \(6\ a\_\(2, 1\)\ \((\(-1\) + 2\ a\_\(2, 1\) + 2\ a\_\(2, \ 2\))\)\)\/\(1 - 6\ a\_\(2, 1\) + 12\ a\_\(2, 1\)\%2 - 6\ a\_\(2, 2\) + 24\ \ a\_\(2, 1\)\ a\_\(2, 2\) + 12\ a\_\(2, 2\)\%2\), b\_1 \[Rule] \(6\ a\_\(2, 1\)\ \((\(-1\) + 2\ a\_\(2, 1\) + 2\ a\_\(2, \ 2\))\)\)\/\(1 - 6\ a\_\(2, 1\) + 12\ a\_\(2, 1\)\%2 - 6\ a\_\(2, 2\) + 24\ \ a\_\(2, 1\)\ a\_\(2, 2\) + 12\ a\_\(2, 2\)\%2\), a\_\(1, 2\) \[Rule] \(\(-1\) + 6\ a\_\(2, 2\) - 12\ a\_\(2, \ 2\)\%2\)\/\(12\ a\_\(2, 1\)\), a\_\(1, 1\) \[Rule] 1\/2\ \((1 - 2\ a\_\(2, 2\))\)}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We still have to fill in values for a[2,1] and a[2,2]. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(coefficients2\ = \ ExtendCoefficients[ coefficients, {a[2, 1]\ \[Rule] \ 1, \ a[2, 2]\ \[Rule] 1}]\)\)\)], "Input"], Cell[BoxData[ \({b\_2 \[Rule] 19\/37, b\_1 \[Rule] 18\/37, a\_\(1, 2\) \[Rule] \(-\(7\/12\)\), a\_\(1, 1\) \[Rule] \(-\(1\/2\)\), a\_\(2, 1\) \[Rule] 1, a\_\(2, 2\) \[Rule] 1}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The generic 2-stage implicit Runge-Kutta method has the following form. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(meth\ = \ ImplicitMethod[2]\)], "Input"], Cell[BoxData[ \({k[1] == x0 + h\ \((a\_\(1, 1\)\ f[k[1]] + a\_\(1, 2\)\ f[k[2]])\), k[2] == x0 + h\ \((a\_\(2, 1\)\ f[k[1]] + a\_\(2, 2\)\ f[k[2]])\), x1 == x0 + h\ \((b\_1\ f[k[1]] + b\_2\ f[k[2]])\)}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["We substitute the coefficients into this method.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(meth2\ = \ meth\ /. \ coefficients2\)], "Input"], Cell[BoxData[ \({k[1] == x0 + h\ \((\(-\(1\/2\)\)\ f[k[1]] - 7\/12\ f[k[2]])\), k[2] == x0 + h\ \((f[k[1]] + f[k[2]])\), x1 == x0 + h\ \((18\/37\ f[k[1]] + 19\/37\ f[k[2]])\)}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We also substitute a concrete choice for the function f and stepsize h. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(meth3\ = \ meth2\ /. \ {\ f[x_]\ -> \ 1\ + x^2 - x^3, \ h\ -> \ 0.01}\)], "Input"], Cell[BoxData[ \({k[1] == x0 + 0.01`\ \((1\/2\ \((\(-1\) - k[1]\^2 + k[1]\^3)\) - 7\/12\ \((1 + k[2]\^2 - k[2]\^3)\))\), k[2] == x0 + 0.01`\ \((2 + k[1]\^2 - k[1]\^3 + k[2]\^2 - k[2]\^3)\), x1 == x0 + 0.01`\ \((18\/37\ \((1 + k[1]\^2 - k[1]\^3)\) + 19\/37\ \((1 + k[2]\^2 - k[2]\^3)\))\)}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We will now perform 500 time steps with this method. I.e.we integrate from x=0 to x=5. 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0.0092221}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]] }, Open ]], Cell["Stability regions", "Subtitle"], Cell["\<\ We will now make a plot of the stability region of the implicit method that \ has just been derived. To that end, we fill in the function f(x)=x.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(meth3\ = \ meth2\ /. \ {\ f[x_]\ \[Rule] \ x}\)\)\)], "Input"], Cell[BoxData[ \({k[1] == x0 + h\ \((\(-\(k[1]\/2\)\) - \(7\ k[2]\)\/12)\), k[2] == x0 + h\ \((k[1] + k[2])\), x1 == x0 + h\ \((\(18\ k[1]\)\/37 + \(19\ k[2]\)\/37)\)}\)], "Output"] }, Closed]], Cell["\<\ The following command needs to be executed in order to load the plotting. \ \>", "Text"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(<< \ Graphics`ImplicitPlot`\)\)\)], "Input"], Cell["\<\ We compute the approximation to x(1) according to this method.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(x1approx\ = \ Simplify[\((x1/x0)\)\ /. \ First[Solve[meth3, \ {x1, \ k[1], k[2]}]]]\)\)\)], "Input"], Cell[BoxData[ \(\(12 + 6\ h + h\^2\)\/\(12 - 6\ h + h\^2\)\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(x1approxabs\ = \ 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