16.920J/SMA 5212 Numerical Methods for PDEs IMPLICIT TIME-MARCHING SCHEME Thus far, we have presented examples of explicit time-marching methods and these may be used to integrate weakly stiff equations Implicit methods are usually employed to integrate very stiff ODEs efficiently. However, use of implicit schemes requires solution of a set of simultaneous algebraic equations at each time-step(i.e. matrix inversion), whilst updating the variables at the same time Implicit schemes applied to OdEs that are inherently stable will be unconditionally stable or A-stable Slide 44 IMPLICIT TIME-MARCHING SCHEME Euler-Backward Consider the euler-backward scheme for time discretization dt Applying the above to the modal equation for parabolic PDe eld (1-h)11-=ahen Slide 45 IMPLICIT TIME-MARCHING SCHEME Euler-Backward Applying the S operator, [(1-h)s-116.920J/SMA 5212 Numerical Methods for PDEs 29 IMPLICIT TIME-MARCHING SCHEME Thus far, we have presented examples of explicit time-marching methods and these may be used to integrate weakly stiff equations. Implicit methods are usually employed to integrate very stiff ODEs efficiently. However, use of implicit schemes requires solution of a set of simultaneous algebraic equations at each time-step (i.e. matrix inversion), whilst updating the variables at the same time. Implicit schemes applied to ODEs that are inherently stable will be unconditionally stable or A-stable. Slide 44 IMPLICIT TIME-MARCHING SCHEME Euler-Backward Consider the Euler-backward scheme for time discretization 1 1 n n n du u u dt h + + − ￾ ✁ = ✂ ✄ ☎ ✆ Applying the above to the modal equation for parabolic PDE du t u ae dt µ = λ + yields ( ) ( ) ( ) 1 1 1 1 1 1 n n n n h n n n h u u u ae h h u u ahe µ µ λ λ + + + + + − ✝ ✞ = + ✟ ✠ − − = Slide 45 IMPLICIT TIME-MARCHING SCHEME Euler-Backward Applying the S operator, ( ) ( 1) 1 1 n n h h S u ahe µ λ + ✡ ☛ − − = ☞ ✌