Backward Euler method yn+1=yn+h,f(t n+1:yn+1 Yn+=y+hayn C Implicit method y(t)= system
Backward Euler Method ( , ) n+1 = n + n n+1 n+1 y y h f t y n n n n n n y h y y h y − + = + + = 1 1 1 1 y' =y Implicit method = ( ) ( ) ( ) 1 z t z t y t d system
The runge-Kutta Methods K,=hf(tn,yn) k 2=hf(tn+ah, yn+ Bk,) yn+1=yn+ak+bk, a+b=1 y(tn+1)-yn+1=O(h3) 2bc=1 2bB=1 k,=hf(t,,yn) RK2 k2=hf(t +h,yn+k,) n+1=yn+(k1+k2)/2
The Runge-Kutta Methods 1 1 2 2 1 1 ( , ) ( , ) y y ak bk k hf t h y k k hf t y n n n n n n = + + = + + = + 2 1 2 1 1 = = + = b b a b ( ) ( ) 3 y t n+1 − yn+1 = O h ( )/ 2 ( , ) ( , ) 1 1 2 2 1 1 y y k k k hf t h y k k hf t y n n n n n n = + + = + + = + RK2: