Solving scalar IVP s : Runge-Kutta Methods Josh Engwer Texas Tech University March 7, NOTATION: h step size x n xt) t n+ t + h x n+ xt n+ ) xt + h) dx = ft, x) SCALAR IVP ASSUMED THROUGHOUT: dt xt ) = x EXPLICIT ν-stage RUNGE-KUTTA METHOD: F = hft n, x n ) F = hft n + c h, x n + a F F = hft n + c h, x n + a F + a F ) F ν = hft n + c ν h, x n + a ν F + a ν F + + a ν,ν F ν ) x n+ = x n + w F + w F + + w ν F ν ) TEST IVP: { x = λx x) = } c a c a a c ν a ν a ν a ν,ν w w w ν w ν x = e λt IMPLICIT ν-stage RUNGE-KUTTA METHOD: F = hft n + c h, x n + a F + a F + + a ν F ν ) F = hft n + c h, x n + a F + a F + + a ν F ν ) F = hft n + c h, x n + a F + a F + + a ν F ν ) F ν = hft n + c ν h, x n + a ν F + a ν F + + a νν F ν ) x n+ = x n + w F + w F + + w ν F ν ) c a a a ν c a a a ν c a a a ν c ν a ν a ν a νν w w w ν c A w T REMARKS: Like Taylor Series Methods, RK-Methods are single step methods and have a high-order truncation error Unlike Taylor Series Methods, RK-Methods do not take derivatives, which would require symbolic programming In the Butcher tableau form, matrix A is strictly lower triangular and c = in explicit RK-Methods Implicit RK-Methods are better suited for stiff IVP s than explicit RK-Methods A unique implicit A-stable ν-stage RK-Method of order ν exists for each positive integer ν A k th -order explicit RK-Method, when applied to Test IVP, gives the k th -order Taylor approximation to e hλ e z A k th -order implicit RK-Method, when applied to Test IVP, gives a k th -order Padé approximation to e hλ e z th -order explicit RK-Methods are ideal as higher-order explicit RK-Methods require more stages as shown here: Order desired 5 6 7 8 Stages required or 5 6 7 or 8 9 or Copyright Josh Engwer Revised March 7,
st -ORDER RUNGE-KUTTA METHODS: F = hft Explicit) Euler s Method : x n+ = x n + hft n, x n ) n, x n ) x n+ = x n + F F = hft Implicit Euler s Method : x n+ = x n + hft n+, x n+ ) n + h, x n + F ) x n+ = x n + F nd -ORDER RUNGE-KUTTA METHODS: Heun s Method : Midpoint Method : Ralston s Method : F = hft n, x n ) F = hft n + h, x n + F ) x n+ = x n + F + F ) Trapezoidal Method : Lobatto IIIC) : A-stable RK : F = hft n, x n ) F = hft n + h, x n + F ) x n+ = x n + F F = hft n, x n ) F = hft n + h, x n + F ) x n+ = x n + F + F ) F = hft n, x n ) F = hft n + h, x n + F + F ) x n+ = x n + F + F ) F = hft n, x n + F F ) F = hft n + h, x n + F + F ) x n+ = x n + F + F ) { F = hft n + h, x n + F ) x n+ = x n + F } rd -ORDER RUNGE-KUTTA METHODS: F = hft n + h, x n + 5 F F ) Radau IIA) : F = hft n + h, x n + F + F ) x n+ = x n + F + F ) 5 th -ORDER RUNGE-KUTTA METHODS: F = hft n, x n ) F = hft n + h, x n + F ) Classic RK : F = hft n + h, x n + F ) F = hft n + h, x n + F ) x n+ = x n + F 6 + F + F + F ) 6 6 F = hft n, x n + F 6 F + F 6 ) F Lobatto IIIC) : = hft n + h, x n + F 6 + 5 F F 6 5 ) F = hft n + h, x n + F 6 + F + F 6 6 ) x n+ = x n + F 6 6 + F + F ) 6 F = hft n + + ] h, x 6 n + F + + ] F 6 ) A-stable RK : F = hft n + ] h, x 6 n + ] F 6 + F ) x n+ = x n + F + F ) 6 + 6 6 6 6 6 ) + ) 6 Copyright Josh Engwer Revised March 7,
EXAMPLE: Show that Ralston s method is a nd -order Runge-Kutta method F = hft n, x n ) F = hft n + h, x n + F ) x n+ = x n + F + F { x = λx x) = F = hλx n F = hλ x n + ] hλx n) = hλx n + hλ) x n hλx n + ] hλ) x n }, which we know the analytic solution is x = e λt x n+ = x n + hλx n] + x n+ = + hλ +! ] hλ) x n = x n + hλx n) + hλx n) + hλ) x n Let z := hλ C Then : x n+ = + z +! ] z x n Kz)x n Notice that e z = Kz) + Oz ) ie Kz) is the nd -order Taylor approximation of e z ] Therefore, Ralston s method is of order = EXAMPLE: Show that Implicit Euler is a st -order Runge-Kutta method F = hft n + h, x n + F ) x x n+ = x n + F n+ = x n + hft n+, x n+ ) x = λx, which we know the analytic solution is x = e x) = λt x n+ = x n + hλx n + F ) = x n + hλx n+ hλ)x n+ = x n x n+ = Let z := hλ C Then : ) x n+ = x n Kz)x n z Notice that Kz) is the, )-Padé approximation of e z Therefore, Implicit Euler method is of order + = ) x n hλ Padé approximations to e z m\n z z + z + z 6 + z + z z 6 z + z + z + 6 + z + z + 6z + z z 6 z 6z + z Copyright Josh Engwer Revised March 7,
EXAMPLE: Show that A-stable RK is a nd -order Runge-Kutta method { F = hft n + h, x n + F } ) x n+ = x n + F First, observe that x n+ = x n + F F = x n+ x n ) F = hf t n + h, x n + ) x n+ x n ) x n+ = x n + hf t n + h, x n + ) x n+ x = λx, which we know the analytic solution is x = e x) = λt x n+ = x n + hλ x n + ) x n+ x n+ = x n + hλx n + hλx n+ ) hλ x n+ = + ) + hλ x n x n+ = hλ ) hλ x n Let z := hλ C Then : + x n+ = z z ) x n x n+ = ) + z x n Kz)x n z Notice that Kz) is the, )-Padé approximation to e z Therefore, A-stable RK method is of order + = Padé approximations to e z m\n z z + z + z 6 + z + z z 6 z + z + z + 6 + z + z + 6z + z z 6 z 6z + z Copyright Josh Engwer Revised March 7,
EXAMPLE: Show that Radau IIA) is a rd -order Runge-Kutta method F = hft n + h, x n + 5 5 F F ) F = hft n + h, x n + F + F ) x n+ = x n + F + F First, observe that F = hft n+, x n+ ) F = hf Now, x n+ = x n + F + F = x n + F + hft n+, x n+ ) F = x n+ x n hft n+, x n+ ) t n + h, x n + 5 F ) hft n+, x n+ ) F = x n+ x n hft n+, x n+ ) 5 F = 5 9 x n+ 5 9 x n 5 6 hft n+, x n+ ) x n + 5 F 5 F = x n + 9 x n+ 5 9 x n 5 ] 6 hft n+, x n+ ) hft n+, x n+ )] x n + 5 F F = 9 x n + 5 9 x n+ 9 hft n+, x n+ ) Hence, F = hf t n + h, 9 x n + 5 9 x n+ ) 9 hft n+, x n+ ) x n+ = x n + t hf n + h, 9 x n + 5 9 x n+ ) 9 hft n+, x n+ ) + hft n+, x n+ ) x = λx, which we know the analytic solution is x = e x) = λt x n+ = x n + t hf n + h, 9 x n + 5 9 x n+ ) 9 hft n+, x n+ ) + hft n+, x n+ ) ) x n+ = x n + hλ 9 x n + 5 9 x n+ 9 hλx n+ + hλx n+ x n+ = x n + hλx n + 5 hλx n+ 6 hλ) x n+ + hλx n+ x n+ = x n + hλx n + hλx n+ 6 hλ) x n+ hλ) + 6 ] hλ) x n+ = + ] hλ) + x n x n+ = hλ) ] hλ) + x n ] 6 hλ) 6 + z Let z := hλ C Then, x n+ = x 6 z + z n Kz)x n Notice that Kz) is the, )-Padé approximation to e z Therefore, Radau IIA) is of order + = Padé approximations to e z m\n z z + z + z 6 + z + z z 6 z + z + z + 6 + z + z + 6z + z z 6 z 6z + z Copyright Josh Engwer Revised March 7,
References ] A Ackleh, E Allen, R B Kearfott, P Seshaiyer, Numerical Analysis: Theory, Methods, and Practice, CRC Press, 9 ] K Burrage and J C Butcher, Stability Criteria for Implicit Runge-Kutta Methods, SIAM Journal on Numerical Analysis, Vol 6, No Feb, 979), pp 6-57 ] JC Butcher, Numerical Methods for Ordinary Differential Equations Wiley & Sons, ] J C Butcher 7) Runge-Kutta methods Scholarpedia, 9):7 5] D Kincaid, W Cheney, Numerical Analysis: Mathematics of Scientific Computing Brooks Cole, Pacific Grove, CA, rd Edition, Copyright Josh Engwer Revised March 7,