TR/8. OCTOBER 97. CUBIC SPLINE INTERPOLATION OF HARMONIC FUNCTIONS BY N. PAPAMICHAEL and J.R. WHITEMAN.
W960748
ABSTRACT It s sown tat for te two dmensonal Laplace equaton a unvarate cubc splne approxmaton n eter space drecton togeter wt a dfference approxmaton n te oter leads to te well-known nne-pont fnte-dfference formula. For armonc problems defned n rectangular regons ts property provdes a means of determnng wt ease accurate approxmatons at any pont n te regon.
.. Introducton Te use of splne approxmatons for partal dfferental equatons and ter relatonsp to fnte-dfference scemes s consdered by Hoskns (970) and Saka (970). Hoskns consders te two dmensonal Posson equaton and sows tat a bvarate cubc Splne approxmaton leads to a nne-pont dfference formula In te more general work of Saka multdmensonal cardnal splnes are used for te approxmaton of varous ellptc and parabolc partal dfferental equatons. In partcular Saka sows tat s splne approxmaton for te two dmensonal Posson equaton leads to te nne-pont dfference formula of Brkoff Scultz and Varga (968). For te eat conducton equaton n one space dmenson te Saka splne approxmaton produces a partcular case of te dfference sceme obtaned by Papamcael and Wteman (97) n wc a cubc splne approxmaton for te space dervatve s combned wt a dfference approxmaton for te tme dervatve. In te present paper a tecnque smlar to te above of Papamcael and Wteman for eat conducton problems s developed for armonc problems n rectangular domans. In ts te doman s covered wt a square mes and t s sown tat for te two dmensonal Laplace equaton a unvarate cubc splne approxmaton n eter space drecton togeter wt a standard central dfference approxmaton n te oter leads to te well-known nne-pont dfference formula. Soluton of te resultng lnear dfference system produces a numercal approxmaton at eac of te grd ponts. Te cubc splne/dfference replacement of te Laplace equaton s
ten used to construct a doubly cubc splne wc nterpolates te soluton of te dscretsed armonc problem. We remark tat te parameters wc determne ts splne are gven at once n terms of known values at te grd ponts. Tus once te fntedfference soluton at te mes ponts s determned te tecnque does not requre te soluton of any oter lnear system. Interpolaton of armonc functons s usually requred wen conformal transformaton metods are used to solve numercally armonc boundary value problems wc ave curved boundares and/or contan boundary sngulartes. In partcular te cubc splne tecnque descrbed n te present paper may be used n conuncton wt a conformal transformaton metod wc maps te doman R of te problem onto a rectangle R. In some cases te transformed armonc problem n R as a smple analytc soluton wc s determned by nspecton; see Wteman and Papamcael (97). In general owever a standard fnte-dfference tecnque s used to determne te soluton of te transformed problem at te grd ponts of a fnte-dfference mes coverng R. Te fnal soluton wc s of course requred at partcular ponts of R s ten obtaned by nterpolaton between te known values at te mes ponts of R. An example on te use of te cubc splne tecnque n conuncton wt a conformal transformaton metod s gven n Secton 5.. Drclet Problems To establs te relatonsp between te cubc splne/dfference approxmaton of Laplace's equaton and te nne-pont dfference.
4. formula we consder te armonc Drclet problem Δ u(x 0 (x ε R () u(x f(x (x ε R In () Δ s te Laplacan operator R s an open doman wt a rectangular boundary R so tat R R {(x: 0 x a 0 y b} a and b beng postve ntegers and f(x s a gven functon contnuous on R. In order to dscretse te problem we cover RU R wt te square mes (x y ) () 0 -------- n ; 0...m were n a and m b and let U be an approxmaton to u(x at te pont (x.y.). We denote by S (x) te cubc splne nterpo- latng te values of U.. at te t...m - mes row and approxmate Laplace's equaton at te ponts (x y ) by M (U U U ) ()... n- ;.... m - were M.. s (x.). We assume tat () s also satsfed wen 0n and tus take M (f f f ) () 0n ;...m - Were f f(x Y ).
5. For te t... m - row te results of Alberg Wlson and Wals (967) sow tat S (x) M (x x) 6 M (x x ) 6 (U M 6 ) (x x) (x x ) ( U M ) (4) 6 x - x x.... n. Hence ' S (x U U ) M M (5) 6 0... n ' S (x U U ) M M (6) 6... n so tat contnuty of te frst dervatves mples U U U M M M 6 6... n-. (7) Te elmnaton of te M' s n (7) by means of equaton ()
6. gves te fnte-dfference equaton D U 0... n - ;... m - (8) were D s te nne-pont dfference operator defned by D a(x a{x- 4a(x a(x 4a(x- - 0a(x 4a(x a(x-y-) 4a(xy-) a(xy-). We now let T ( be te cubc splne nterpolatng te values U at te t -...n - mes column and approxmate Laplace's equaton at (x y ) by N (U U U ) (9)... n ;.... m were N T" (y.). Ten correspondng to () (4) and (7) respectvely we ave N (f f f ) (0)... n - ; 0 m
7. T ( N (y 6 N (y y ) 6 (U 6 N ) y y (U 6 N ) y y () Y Y Y J m and N 6 N N 6 IJ U U U ()... m-. Te elmnaton of te N's n () by means of (9) gves agan te equaton (8). It s tus sown tat te cubc splne/dfference replacements () and (9) to te Laplace equaton bot produce te well-known nne-pont dfference approxmaton (8) wc as local truncaton error of order 6 ; see e.g. Forsyte and Wasow (960 p.94).. Cubc Splne Interpolaton. Te applcaton of (8) at eac nternal mes pont togeter wt te boundary condtons of () leads to a postve defnte dagonally domnant lnear system of {(n-l)x(m-l)} equatons wc s solved for te unknowns U (l)n - (l)m -. Formulae ()- (U) and (9)- () ten produce te cubc splnes S.(x) and T.( wc approxmate respectvely te soluton u(xy )
8. at te t mes row and te soluton u(x at te t mes column. We note tat D S (x) D U (xy ) 0 () x - x x... n - ;... m - and (4) D T ( D U(x 0 y - y y... n- ;.m- Tese follow at once from ( 4 ) () and (8) snce D M - / {D U - - D U D U } 0 and.. n - ;.. m - D N - / {D U - - D U D U } 0...n- ;. m -. We now descrbe a metod for nterpolatng te soluton of () at any pont (xε R. For ts we let r be te square r {(x: x - x x y - y y }.n ;..m
9. and consder tree procedures eac dealng wt a dfferent part of R. Procedure I Used for (x ε r -...n ;...m - We determne te sx values T - (y k) k - 0 T (y k) and ence usng te approxmatons of Secton we calculate U(x from te cubc splne nterpolatng te values U(x - 0...n. Tus we take (see equaton (4)) U(x Q (x M ( (x x) 6 M ( (x x ) 6 (T ( 6 M () (x x) (T ( 6 M () (x x ) (x ε r were M ( - {T (y-) - T ( T (} wt T 0 ( f(0 and T n ( f(a.
We note tat 0. Q ( xy ) S (x) and Q (x T (. Procedure II. Used for (x ε r...n - ;...m. Te value of U(x s obtaned by ntercangng te roles of x and y n te tecnque of Procedure I. Tus we determne te sx values S - (x k) S (x k) k - 0 and ence calculate U(x from U(x Q (x N (x) (y 6 N (x) (y y ) 6 (S (x) 6 N (x)) (y (y y ) (S (x) N (x)) (x 6 r were N (x) - / {S (x-) - S (x) S (x)} wt S 0 (x) f(x0) and S m (x) f(x b). Agan we note tat Q (x y ) S (x) and Q (x T (.
Te functons Q (x and Q (x are bcubc n eac mes square r of te rectangles R {(x: 0 x a y b - } and R {(x: x a - 0 y b} respectvely. By use of te contnuty propertes of S (x) and T ( t can be sown tat Q (x s n C 4 (R ) and Q (x s n C 4 (R ). (By te termnology of A l be r g et al.(l967 P-5) C n r (R) s te famly of functons F(x on R wose n t order partal dervatves nvolvng no more tan r t order dfferentaton wt respect to a sngle varable exst and are contnuous.) It follows tat Q (x and Q (x are "smple double cubc splnes" nterpolatng te values U at te mes ponts of R and R respectvely; see Alberg et al. (967 p.5-9). Also t can be sown tat Q (x Q (x (x ε R R and by means of () and (4) tat (5) D U(x 0 (x ε r 4... n - ; 4....m -. Procedure III. Used for (x ε r. n ; m. We assume tat (5) olds for (x ε R R and use ts nne-pont formula to express U(x n terms of values tat
. can be nterpolated by means of Procedure or II. Tus f (x (pq) ε r n ; m we take Upq - ( 4U Pα q U pα q ) - (4U pqß - 0U pα q ß 4U pα q ß ) - (U pqß 4U pα qß U pα qß ) were U k l U(kl) and α wen α - wen n β wen and β - wen m. To summarze te above we note tat te value U(x approxmatng te soluton u(x at any pont (x ε R may be calculated by usng (a) Procedure I f (x. ε R (b) Procedure II f (xε r...n - ; m (c) Procedure III f (x ε r n ; m or by usng (a) Procedure II f (x ε R (b) Procedure I f (x ε r n;.... m - (c) Procedure III f (x ε r n ; m. Snce wen (x ε R R Q (x Q (x t s clear tat for any (x ε R bot te above two metods produce te same result. For (x ε R R approxmatons to u(x x may be
. determned from U(x x Q (x x M ( (x x) M ( (x x ) (6) (T ( M () (T ( 6 6 M () f (x ε r r ε R and from. U(x Q x x (x (y N (x) - 6 N (y (x) y ) - 6 (7) (S (x) 6 N (x)) y y (S (x) 6 N (x)) (y y ) f (x ε r - --------------- n - ; m. In (7) N (X) - { S (x-)- S (x) S (x)} and S.(x) s found by dfferentatng (4) wt respect to x. To determne an approxmaton to u( x x at a pont (xε r n ; m we note tat u(x D x 0 (x ε r (8) 4... n - ; 4... m -. we assume tat formula (8) olds for any (xε R R and use t as n Procedure III to express u(x x n terms of values tat
4. can be calculated by means of (6) and (7). Approxmatons to u(x x are determned n a smlar manner from U(x x Q (x y f(x R U(x y Q (x y f (x ε r n ;...m - and by usng U(x D 0 f(x r n; x m. 4. Mxed Boundary Value Problems To llustrate te applcaton of te tecnque to mxed boundary value problems we consder te problem () but on te sde R * {(0: 0 < y < b } of R we replace te Drclet boundary condton by a Neumann condton. Tus we consder te armonc problem Δu(x 0 (xε R u(x g( x (xε R * (9) u(x f(x (xε R - R *
5. were g( s a gven functon contnuous on R *. All approxmatons and results of Secton stll old except () for wc we ave M 0 - (U 0 - - U o U o ) (0) and M n - (f n- - f n f n )... m- were te U o ()m - are not known and must be determned. Te boundary condton (9) s approxmated at te pont (0 y.) by S' (0) g... m - or on usng (5) by M 0 6 M U U 0 g. () Te elmnaton of te M's n () by means of () and (0) gves te fnte-dfference equaton U o U - 0U o 4U U o - U - 6g ()... m - approxmatng te soluton of te problem at te boundary ponts ( 0 y ). Te approxmaton () as local truncaton error of
6. order 4 and s equvalent to usng te dfference approxmaton {U U ) 4(U U ) (U U )} for te dervatve n (9) n conuncton wt te nne-pont formula appled at te pont (0. Te applcaton of () at te (m - ) boundary ponts ( 0 y ) () m - and of te nne-pont formula at te nternal mes ponts leads to a lnear system of nx(m - ) equatons wc s solved for te unknowns U 0()n - ; ()m -. Te tecnque of Secton s ten used to nterpolate U(x at any pont (xε R U R * from te values U at te mes ponts. However for (xε r r ε R te determnaton of U(x from te double cubc splne Q (x requres te knowledge of T o ( U(0 and M o ( - {U(0y-) - U(0 U(0}. Snce T ( s not defned for 0 te unknowns U(0 and M o ( are determned as follows. Te boundary condton (9) (0 by s approxmated at te pont Q (0 x g(
7. or on usng (l6)by M 0 ( 6 M ( T ( U(0 g(. () Equaton () togeter wt 6 M 0 ( M ( M ( U(0 T ( T ( 0 (4) 6 ten gves te two relatons U(0 7 / 6 M ( M ( 5T ( - T ( - g( and M o ( -5 M ( - M ( -6T (7) 6T ( - 6g( wc express te unknowns U(0 and M o ( n terms of values tat can be determned from te cubc splnes T ( and T (. We remark tat (4) wc follows at once from te constructon of Q (x and can be verfed easly by means of (4) s te contnuty relaton wc sows tat Q (x x Q (x x Te applcaton of te tecnque to problems wt Neumann condtons on any of te oter tree sdes of R s clear.
8. 5. Numercal Results Problem (Drclet Problem) Δu(x 0 (xε R u(x cosxsny (xε R were R U R {(x : 0 x 0 y }. A square mes of sze 0. s used. Te fnte-dfference soluton computed by applyng te nne-pont formula at te nteror mes ponts (0. 0.) ()4 ; ()4 s accurate to egt sgnfcant fgures. at te ponts Numercal results obtaned by te cubc splne tecnque ( 0. 0.l) ()9 ; ()9 are gven n Table. Tey are compared wt: () values computed from te analytc soluton and () values obtaned by nterpolatng te results at te mes ponts usng te bvarate nterpolaton formula ~ F(x p y q) q(q ) F p(p ) F ( pq p q )F p(p q ) F q(q p ) F pq F (5) p <. q <
of Abramowtz and Stegun (965 eqn.5..67). Formula (5) 9. wc determnes an approxmaton ~ F (x to F(x n terms of values of F(x at sx grd ponts of a square mes as truncaton error e p q F(x py q) - ~ F (x py q) of order. However wen F(x s armonc and p q ½ te order of te truncaton error rses to 4. In partcular wt F(x u(x cos x sn y and 0. e Ω 4 4 u x y 0.5 0 4 cosx sny wereas e 0 Ω 6 u y 0.5 0 cosx cosy. Te last two equatons explan wy te results obtaned "by te use of (5) at te ponts (0. 0.) () 9 5 () 9 are muc more accurate tan tose obtaned at te ponts (0. O.l) ()4 ; ()9. Problem (Mxed Boundary Value Problem) Au(x 0 (x ε R u(x x cos y (x R * u(x sn x cos y (x ε R - R *
were 0. R* {(0 : 0 < y < } and R U R {(x :0 x ;0 y }. A square mes of sze 0. s used. Te fnte-dfference soluton computed by applyng te nne-pont formula at te nteror mes ponts (0. 0.) ()4 ; (}4 and formula () at te boundary ponts (0 0.) - ()4 s accurate to fve sgnfcant fgures. Numercal results obtaned by te cubc splne tecnque at te ponts (0. 0.) 0()9 ; ()9 are gven n Table and are compared wt values computed from te analytc soluton. To llustrate te use of te cubc splne tecnque n conuncton wt a conformal transformaton metod we consder te followng armonc problem. Problem. Δu(x 0 (xε R u(x (xε R u(x 0 (xε R were R s te sem-crcular open doman R {(x : x y < x < y > 0}
. wt boundary R R U R were R {(x : x y x < y > 0} and R ABC - {(x0) : x < } wt A (0) B (00) and C (-0). Te tree successve conformal transformatons z w w (6) t (7) z and w k( ) sn (t were sn denotes te Jacoban ellptc sne and K(t//) s te complete ellptc ntegral of te frst knd wt modulus / ) (8) are used to map G R U R n te w x y - plane onto te square G R U R n te w ξ n - plane wt {(ξ n) : o ξ } w plane w plane A (00) B (0) C (0).
. Tus te combned effect of (6) (7) and (8) s to transform te orgnal problem n G nto te problem Δv(ξn) 0 (ξn) ε R v(ξn ) ( ξn)ε R (9) v ( n ) 0 (ξn)ε R n G R U R were R R U R and R {(ξ0) : 0 < ξ < } R R - Full computatonal detals for te mplementaton of te above conformal transformaton metod are gven n Wteman and Papamcael (97). Te boundary values of te transformed problems ave ump dscontnutes at te corners (00) and (0) of R. A detaled dscusson on tecnques for removng suc dscontnutes from R te boundary condtons of armonc problem n rectangular regons s gven by Rosser (97). In te present case we ntroduce te armonc functon V( ε n ) v( ε n ) π {arctan( ε η ) arctan( η ε )} (0) and nstead of (9) we consder te problem ΔV(ξn) 0 V(ξ n) - (ξn)ε R' (ξn)ε R () were V( n) f(ξn) (ξn)ε R f( ε η) π {arctan( ε ) η arctan( ε )}. η
Te nne-pont formula s used to determne approxmatons. V ~ to te soluton of () at te mes ponts ( 0. 0. ) () ; () 4 of a square mes of sze 0. coverng G. of te orgnal problem s of course requred n R. Te soluton To determne an approxmaton to ts soluton at a specfc pont P (x ε R P s mapped nto te pont p ε R. Snce p wll not n general be a pont of te fnte-dfference mes coverng G tecnque s used to compute an approxmaton to V( p ) te known values te cubc splne n terms of V ~ at te mes ponts n G. An approxmaton to u(p) v ( p ) s ten found by means of (0). Numercal approxmatons to u(x obtaned by usng te conformal transformaton metod (CTM) n conuncton wt te cubc splne tecnque are gven n Table. gven at te ponts Te results are (0.l 0.) 0()9 ; ()9 of te quadrant { (x : x y < x > 0 y > 0} of R and are compared wt () values computed from te analytc soluton and () u(x arctan{y/( -x -y )} values computed by usng formula (5) to nterpolate te fnte-dfference approxmatons V ~ n G.
We remark tat wen a square mes of sze 0. s used to determne te fnte-dfference soluton of () and to perform te cubc splne nterpolaton n G te approxmatons U(x to u(x computed at te ponts of Table are suc tat u(x - u(x x 0-6.
At eac pont te numbers represent: () Upper entry: Value computed by Splne Interpolaton ()Mddle entry: Value computed from te Analytc Soluton () Lover entry: Value computed by usng Interpolaton Formula 5. y 0.9 0.7 0.5 0..06.00605 0.98066 0.94548 0.900848 0.8477 0.7859 0.7578 0.68074.088.04.006055.0067 0.980669 0.98064 0.754789 0.74459 0.74697 0.754794 0.7446 0.7470 0.75476 0.74405 0.74670 0.58489 0.5849 0.58470 0.0997 0.0999 0.0987 0.09966 0.099666 0.099658 0.50706 0.50708 0.544 0.98449 0.98450 0.98954 0.49787 0.4978 0.4 97800 0.9097 0.9099 0.90908 0.09870 0.09569 0.09870 0.09767 0.09569 0.095685 0.945485 0.946 0.90085 0.9008 0.698699 0.66574 0.69870 0.66570 0.69957 0.665689 0.479959 0.47996 0.480464 0.8048 0.8048 0.80955 0.0959 0.0960 0.0979 0.45700 0.45704 0.45784 0.679 0.674 0.67 0.087904 0.087905 0.087897 0.847 0.84757 0.66084 0.66086 0.6677 0.40077 0.40079 0.40 0.50 0.5 0.5659 0.0867 0.0867 0.0855 0.785 0.785086 0.5809 0.58097 0.58070 0.9855 0.98556 0.9858 0.908 0.90 0.90 0.0766 0.0766 0.076605 0.758 0.75656 0.58508 0.5850 0.5890 0.6049 0.605 0.64 0.60 0.6 0.59 0.069787 0.069787 0.0694 0.6809 0.6806 0.47540 0.4754 0.475 0.96 0.98 0.90 0.899 0.899 0.8985 0.066 0.0665 0.0659 0. 0 0. 0. 0. 0.4 0.5 0.6 0.7 0.8 0.9.0 x TABLE
At eac pont te numbers represent: () Upper entry: Value computed by Splne Interpolaton () Lower entry: Value computed from te Analytc Soluton 0.9 0.7 0.5 0.000000 0.406 0.84708 0.450 0.558067 O.687054 0.80978 0.94.0809.545 0.000000-0.000004 0.4070 0.504 0.8470 0.496 0.4506 0.709 0.558070 0.48878 0.687058 0.60754 0.8098 0.70878 0.90 0.808595.080 0.90099.575 0.980 0.000000-0.000004 0.508 0.57 0.4964 0.40 0.7098 0. 0.488786 0. 495 0.60760 0.540607 0.7087 0.6670 0.80860 0.7640 0.90040 0.808906 0.9808 0.8894 0.000000 0.575 0.405 0.6 0.498 0.5406 0.66706 0.7647 0.808909 0.8800 0. -0.00000 0.000000 0.0457 0.0460 0.07674 0.07677 0.0895 0.0899 0.4 0707 0.407074 0.54057 0.506 0.5904 0.5904 0.6740 0.6746 0.74 9877 0.749880 0.8887 0.8884 0.000000 0.008 0.9966 0.96997 0.965 0.4 88 0.567466 0.64747 0.7094 0.7875 0. 0.000000 0.00 0.99664 0.96999 0.967 0.4 885 0.567468 0.64744 0.70946 0.78747 0 0. 0. 0. 0.4 0.5 0.6 0.7 0.8 0.9.0 TABLE. 6
TABLE At eac pont te numbers were computed from; Y 0.9 0.9048 0.96548 0.947070 0.964668 0.98990 0.9049 0.9845 0.96549 0.96650 0.94707 0.947007 0.964669 0.96495 0.9899 0.988604 () Upper entry: CTM and Cubc Splne Interpolaton. () Mddle entry: Analytc Soluton. 0.7 0.777599 0.777600 0.77774 0.786 0.7864 0.78069 0.7980 0.79804 0.799 0.84450 0.8445 0.847 0.84409 0.880 0.9049 0.990906 0.84404 0.8486 0.880 0.88845 0.9050 0.9798 0.990906 0.99070 () Lover entry: CTM and Interpolaton Formula (5). 0.5 0.5904 0.59448 0.606946 0.686 0.660658 0.70480 0.766 0.88057 0.9049 0.5904 0.59085 0.59448 0.59486 0.606947 0.60697 0.686 0.686 0.660660 0.660775 0.7048 0.705 0.7669 0.7655 0.88064 0.8805 0.905 0.99809 0. 0.709 0.7094 0.70909 0.749 0.744 0.749 0.8454 0.8459 0.847 0.4045 0.4047 0.4097 0.4955 0.4955 0.4965 0.46969 0.469708 0.469804 0.576 0.5766 0.57675 0.667 0.600 0.6050 0.7078 0.7080 0.704 0.894857 0.89486 0.89488 0. 0.6906 0.690 0.6880 0.856 0.86 0.80 0.089 0.096 0.744 0.90 0.909 0.90 0.5059 0.505 0.50448 0.6805 0.68045 0.68 0.95649 0.95695 0.95759 0.478 0.48 0.48 0.045 0.0499 0.0466 0.5457 0.5475 0.587 0.0 0. 0. 0. 0.4 0.5 0.6 0.7 0.8 0.9 7.
8. References. ABRAMOWITZM. and STEGUNI.A.: Handbook of matematcal functons. New York: Dover 965.. AHLBERG J.H. NILSGNE.N. and WALSHJ.L.: Te teory of splnes and ter. applcatons London: Academc Press 967.. BIRKHOFFG. SCHULTZ M.H. and VARGAR.S.: Pecewse Hermte nterpolaton n one and two varables wt applcaton to partal dfferental equatons Numer.Mat. -56 (968). 4. FORSYTHE G.E. and WASOWW.R.: Fnte-dfference metods for partal dfferental equatons. New York: Wley 960. 5. HOSKINS W.D.: Studes n splne approxmaton and varatonal metods. Doctoral Tess Brune Unversty 970. 6. PAPAMICHAEL N. and WHITEMAN J.H.: A cubc splne tecnque for te one dmensonal eat conducton equaton J.Inst.Mats Apples. - (97). 7. ROSSER J.B.: Fnte-dfference soluton of Posson's equaton n rectangles of arbtrary sape. Tecncal Report TR/7 Dept. of Matematcs Brune Unversty 97. 8. SAKAI M. :Multdmensonal cardnal splne functon and ts applcatons. Memors of Faclty of Scences Kyusu Unversty Seres A XXIV 40-46 (970). 9. WHITEMANJ.R. and PAFAMICHAELN.: Treatment of armonc boundary value problems contanng sngulartes by conformal transformaton metods. Z.angev.Mat.Pys. 655-664( 97). NOT TO BE REMOVED FROM THE LIBRARY