P-stable Exponentially fitted Obrechkoff methods

Transcription

1 P-stable Exponentially fitted Obrechkoff methods Marnix Van Daele, G. Vanden Berghe Vakgroep Toegepaste Wiskunde en Informatica Universiteit Gent SciCADE 7 p. /4

2 Outline Introduction on exponentially fitted (EF methods 2-step Obrechkoff methods for y = f(x, y P-stable Obrechkoff methods for y = f(x, y P-stable EF Obrechkoff methods for y = f(x, y Conclusions SciCADE 7 p. 2/4

3 Exponentially fitted methods In the past 5 years, our research group has constructed modified versions of well-known linear multistep methods Runge-Kutta methods Aim : build methods which perform very good when the solution has a known exponential of trigonometric behaviour. SciCADE 7 p. 3/4

4 Linear multistep methods A well known method to solve y = f(y y(a = y a y (a = y a is the Numerov method (order 4 y n+ 2 y n + y n = h2 2 (f(y n + f(y n + f(y n+ Construction : impose L[z(x;h] = for z(x =, x, x 2, x 3, x 4 where L[z(x;h] := z(x + h + α z(x + α z(x h h 2 ( β z (x + h + β z (x + β z (x h SciCADE 7 p. 4/4

5 Exponential fitting Consider the initial value problem y + ω 2 y = g(y y(a = y a y(a = y a. If g(y ω 2 y then y(x α cos(ω x + φ To mimic this oscillatory behaviour, one could replace polynomials by trigonometric (in the complex case : exponential functions. SciCADE 7 p. 5/4

6 EF Numerov method Construction : impose L[z(x; h] = for z(x =, x, x 2, sin(ω x, cos(ω x L[z(x;h] := z(x + h + α z(x + α z(x h h 2 ( β z (x + h + β z (x + β z (x h y n+ 2 y n + y n = h 2 (λf(y n + ( 2 λ f(y n + λf(y n+ λ = 4 sin = := ω h SciCADE 7 p. 6/4

7 EF methods Generalisation : to determine the coefficients of a method, we impose conditions on a linear functional. These conditions are related to polynomials : {x q q =,..., K} exponential or trigonometric functions, multiplied with powers of x : {x q exp(±µx q =,..., P } or, with ω = i µ, {x q cos(ωx, x q sin(ωx q =,..., P } Classical method : P = SciCADE 7 p. 7/4

8 Choice of ω local optimization based on local truncation error (lte ω is step-dependent global optimization Preservation of geometric properties (periodicity, energy,... ω is constant over the interval of integration SciCADE 7 p. 8/4

9 Obrechkoff methods Nikola Obrechkoff ( Obrechkoff methods (OM : 94 for quadrature Milne : OM for solving diff. eq. : 949 SciCADE 7 p. 9/4

10 Obrechkoff methods for y = f(x, y y n+ 2y n + y n = Two-step methods m i= ( h 2 i β i y (2i n+ + 2β i y n (2i + β i y (2i n L[z(x;h] := z(x + h 2 z(x + z(x h m ( h 2 i β i z (2i (x + h + 2β i z (2i (x + β i z (2i (x h i= symmetric method : L[z(x; h] if z(x is odd L[;h] order p L[x q ;h] =, q =,,..., p + lte = C p+2 h p+2 y (p+2 (x n + O ( h p+3 C p+2 = L[xp+2 ;h] (p + 2!h p+2 SciCADE 7 p. /4

11 Two-step OM m = : p = 4, C 6 = 24 m = 2 : p = 8, C = y n+ 2y n + y n = h 2 2 (Numerov method ( y (2 n+ + y(2 n + y (2 n y n+ 2y n + y n = h h4 52 ( y (2 n+ + 23y(2 n + y (2 n ( 3 y (4 n+ 626 y(4 n + 3y (4 n SciCADE 7 p. /4

12 Two-step OM m = 3 : p = 2, C 4 = y n+ 2y n + y n = h h h ( 229 y (2 n y(2 n + 229y (2 n ( y (4 n+ method of order p = 4m 422 y(4 n + y (4 n ( 27 y (6 n y(6 n + 27y (6 n SciCADE 7 p. 2/4

13 Stability of two-step OM y n+ 2 y n + y n = m i= ( h 2 i β i y (2i n+ + 2β i y n (2i + β i y (2i n applied to y = λ 2 y gives y n+ 2R mm (ν 2 y n + y n = ν := λh m + ( i β i ν 2i R mm (ν 2 = + i= m ( i+ β i ν 2i i= The stabilitity function R mm uniquely determines the method. A method has the interval of periodicity (, ν 2 if R mm (ν 2 < for < ν 2 < ν 2. The method is P -stable if R mm (ν 2 for all real ν. SciCADE 7 p. 3/4

14 Stability of two-step OM m = : p = 4, C 6 = 24 (Numerov ν2 = 6 y n+ 2y n + y n = h 2 2 ( y (2 n+ + y(2 n + y (2 n R 6 ν 2 SciCADE 7 p. 4/4

15 Stability of two-step OM m = 2 : p = 8, C = y n+ 2y n + y n = h h ( y (2 n+ + 5y(2 n + y (2 n ( 3 y (4 n+ 626 y(4 n + 3y (4 n+ ν 2 = 25.2 R ν 2 SciCADE 7 p. 5/4

16 Stability of two-step OM m = 3 : p = 2, C 4 = y n+ 2y n + y n = h h h ( 229 y (2 n y(2 n + 229y (2 n ( y (4 n y(4 n y (4 n ν 2 = ( 27 y (6 n y(6 n + 27y (6 n R ν 2 SciCADE 7 p. 6/4

17 P-stable 2-step OM Ananthakrishnaiah (987 idea : use some parameters to obtain P-stability. y n+ 2y n + y n = m = 3 and m = 4 e.g. m = 3 h 2 ( β y (2 n+ + 2β y (2 n + β y (2 n +h 4 ( β 2 y (4 n+ + 2β 2 y (4 n + β 2 y (4 n +h 6 ( β 3 y (6 n+ + 2β 3 y (6 n + β 3 y (6 n impose p = 6 : {x 2, x 4, x 6 }, then 3 parameters (β 2, β 3 and β 3 remain R 33 = β ν 2 + β 2 ν 4 β 3 ν 6 + β ν 2 β 2 ν 4 + β 3 ν 6 let β 3 = β 3 SciCADE 7 p. 7/4

18 Ananthakrishnaiah s idea Choose these 2 remaining parameters β 2, β 3 such that the method R 33 = N 3 D 3 becomes P-stable < (D 3 N 3 (D 3 + N 3 >.4 β 2.8. β 3 SciCADE 7 p. 8/4

19 Ananthakrishnaiah s idea Find the solution for which the phase-lag, ( 3 ν arccos R 33 = β β 2 ν 7 + O ( ν 9.4 becomes minimal. β 2.8. β 3 SciCADE 7 p. 9/4

20 Ananthakrishnaiah s idea Find the solution for which the phase-lag, ( 3 ν arccos R 33 = β β 2 ν 7 + O ( ν 9.4 becomes minimal. β 2.8. This gives (β 3, β 2 = β 3 ( 44, 6. SciCADE 7 p. 2/4

21 Ananthakrishnaiah s idea Ananthakrishnaiah m = 3 : p = 6, P-stable, C 8 = 54 y n+ 2y n + y n = h 2 2 h4 6 + h6 44 ( y (2 n+ + 8y(2 n + y (2 n ( y (4 n+ 22 y(4 n + y (4 n ( y (6 n+ + 2y(6 n + y (6 n SciCADE 7 p. 2/4

22 Ananthakrishnaiah s idea R 33 = 9 2 ν2 + 6 ν4 44 ν6 + 2 ν2 + 6 ν ν6 R R 33 = N 3 D ν < since (D 3 N 3 (D 3 + N 3 = ν2 (ν 2 2 (ν SciCADE 7 p. 22/4

23 P-stable 2-step OM We were able to generalise Ananthakrishnaiah s idea in M. Van Daele and G. Vanden Berghe, P-stable Obrechkoff methods of arbitrary order for second-order differential equations, Numerical Algorithms 44, 27, 5-3 Algoritm to construct a P-stable OM for a given m : impose order 2 m D m N m and D m + N m should both be halves of perfect squares This leads to a system of non-linear equations. Theorem : the approximant R mm (ν 2, obtained by generalising Ananthakrishnaiah s approach, is given by the real part of the (m, m-padé approximant of exp(i ν. This leads to a system of linear equations. SciCADE 7 p. 23/4

24 Example R 33 (ν 2 = 9 2 ν2 + 6 ν4 44 ν6 + 2 ν2 + 6 ν ( ν6 + 2 = R i ν ν2 2 i ν3 2 i ν = ( ν2 + 2 i ν3 ν2 2 ( 2 ν 2 ν3 2 ( ν2 2 ( + 2 ν 2 ν3 2 where + 2 ν + ν2 + 2 ν3 2 ν + ν2 = exp(ν + O ( ν 7 2 ν3 ( R 33 (ν 2 = ν2 2 ( 2 ν 2 ν3 2 ( ν2 2 + ( 2 ν 2 ν3 2 SciCADE 7 p. 24/4

25 Conclusion How does the P-stable Obrechkoff method m ( y n+ 2y n + y n = h 2 i β i y (2i n+ + 2β i y n (2i + β i y (2i n i= of order 2 m look like? i β i = ( i+ a 2 i + 2 ( j+ a j a 2i j β i = a 2 i + 2 i j= where a j = j= a j a 2i j ( m j ( 2 m j for j m for j > m i =..., m SciCADE 7 p. 25/4

26 P-stable Expon. fitted OM How to obtain P-stable exponentially-fitted Obrechkoff methods? m ( y n+ 2 y n + y n = h 2 i β i y (2i n+ + 2β i y n (2i + β i y (2i n i= applied to y = λ 2 y gives y n+ 2R mm (, ν 2 y n + y n = with := ω h and ν := λh Padé-approximants = exponentially-fitted Padé approximants Construction of Polynomially-fitted (m, m Padé approximants : P m (x P m ( x = exp(x + O ( x 2 m+ exp(x P m ( x P m (x = O ( x 2 m+ d2 q dx 2 q (exp(x P m( x P m (x x= = q =,..., m SciCADE 7 p. 26/4

27 EF Padé approximants F(x, t = exp(t x V m ( t x V m (t x V m (x = + 2q x 2q F(x, t (x,t=(, = q =,..., K ( R q t q F(x, t = q =,..., P (x,t=(i, where K m and P + K + = m. m a i x i j= This leads to a system of m linear equations in the unknowns a i, i =,..., m. The EF (K, P Padé approximant to exp(ν is then given by (K,PˆP m m (ν = V m (ν/v m ( ν. SciCADE 7 p. 27/4

28 EF Padé approximants : m = F(x, t = exp(t x V ( t x V (t x V m (x = + a x (K =, P = 2 x 2 F(x, t (x,t=(, = 2 ( 2 a = a = 2 (, ˆP (ν = + 2 x (K =, P = 2 x ( R (F(i, = R e i ( i a ( + i a = a = sin (cos + = 2 tan(/2 /2 (,ˆP (ν = tan(/2 /2 x tan(/2 /2 x SciCADE 7 p. 28/4

29 EF Padé approximants : m = m = (K, P a (, (, 2 tan 2 SciCADE 7 p. 29/4

30 EF Padé approximants : m = m = (K, P a (, (, O(4 SciCADE 7 p. 3/4

31 EF Padé approximants : m = m = (K, P a (, a.5.5 (, a 2π 4π 6π 8π π.5.5 2π 4π 6π 8π π SciCADE 7 p. 3/4

32 EF Padé approximants : m = 2 m = 2 (K, P a a 2 (2, (, (, ( cos ( + sin 2 2 tan tan 2 sin ( + sin 2 SciCADE 7 p. 32/4

33 EF Padé approximants : m = 2 m = 2 (K, P a a 2 (2, (, (, O( O ( O(4 SciCADE 7 p. 33/4

34 EF Padé approximants : m = 2 m = 2 (K, P a a 2 (2, a.5 a π 4π 6π 8π π. 2π 4π 6π 8π π (, a.5 a (, a 2π 4π 6π 8π π π 4π 6π 8π π a 2 2π 4π 6π 8π π π 4π 6π 8π π SciCADE 7 p. 34/4

35 m = 3 (K, P a a 2 a 3 (3, a.5 a 2..5 a π 4π 6π 8π π.5. 2π 4π 6π 8π π π 4π 6π 8π π.5 (2, a.5 a 2..5 a (, a 2π 4π 6π 8π π.5 a π 4π 6π 8π π a 3.5 2π 4π 6π 8π π π 4π 6π 8π π 2π 4π 6π 8π π.5 2π 4π 6π 8π π.5 (,2 a.5 a 2..5 a π 4π 6π 8π π 2π 4π 6π 8π π.5 2π 4π 6π 8π π SciCADE 7 p. 35/4

36 The Stiefel-Bettis problem z + z =. exp(ix z( = z ( =.995i x 4π equivalent real form u + u =. cos(x, u( =, u ( =, v + v =. sin(x, v( =, v ( = z(x = u(x + i v(x u(x = cos(x +.5x sin(x v(x = sin(x.5x cos(x γ(x = u 2 (x + v 2 (x = + (.5x 2 γ(4 π h = 2 i π ω = SciCADE 7 p. 36/4

37 The Stiefel-Bettis problem m = log(abs. err. in γ(4π log(h : P = : P = SciCADE 7 p. 37/4

38 The Stiefel-Bettis problem m = 2 log(abs. err. in γ(4π log(h : P = : P = : P = SciCADE 7 p. 38/4

39 The Stiefel-Bettis problem m = log(abs. err. in γ(4π log(h : P = : P = : P = : P = 2 SciCADE 7 p. 39/4

40 Conclusion two-step P-stable exponentially fitted Obrechkoff methods : for any given m, P-stable EF (K, P-methods of order 2 m exist the construction is based on EF Padé approximants to the exponential function the coefficients depend on a parameter the coefficients are continous functions of iff P is odd numerical example was given to illustrate the theory SciCADE 7 p. 4/4