Coupled ODEs to Stability Analysis Uncoupled ODEs Starting from du= au+b dt Premultiplication by e yields du E Au+e b da E EA(EE- Ju+E-b du E E- AEEu+E b SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS 11 Stability Analysis Coupled ODEs to Uncoupled ODEs Starting from du Au b dt = + G G G 1 Premultiplication by E yields − 1 du 1 E E Au E b dt − − = G G G ( 1 du 1 1 E EA EE u E b dt − − − = G G G I ( 1 du 1 1 E E AE E u E b dt − − − = G G G Λ −1 + ) −1 + ) −1 +
Coupled ODEs to Stability Analysis Uncoupled ODES Continuing from du E -AEu+e b Let u=e-u andf=e b we have U=AU+F which is a set of Uncoupled ODES. SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS 12 Stability Analysis Coupled ODEs to Uncoupled ODEs 1 Let UE uand F E b, we have − = G G G d U F dt = Λ + G JG JG which is a set of Uncoupled ODEs! 1 du 1 E u E b dt − − = Λ + G G G Continuing from −1 = G U 1 E −
Coupled ODEs to Stability Analysis Uncoupled ODES Expanding yields dU1 17+F du 1U2+F2 L=nU. +F dU UN+FN-1 Ince the equations are independent of one another, they can be solved separately y The idea then is to solve for u and determine u= eu SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS 13 Stability Analysis Coupled ODEs to Uncoupled ODEs Expanding yields Since the equations are independent of one another, they can be solved separately. The idea then is to solve for U and determine = EU G G G 1 11 1 dU U dt = λ + 2 2 2 dU U dt = λ + j j j dU U dt = λ + 1 1 1 N N N dU U dt λ − = − − u F 2 F j F N 1 + F −
Coupled ODEs to Stability Analysis Uncoupled ODES Considering the case of b independent of time for the general i th equation s the solution for j=1, 2,..N-1 Evaluating, u=EU=E(ceEA Eb Complementary Particular(steady-state ( transient) solution solution where ce=celt SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS 14 Considering the case of independent of time, for the general equation, th b j G Stability Analysis Coupled ODEs to Uncoupled ODEs jt 1 j j j U e F λ λ = is the solution for j = 1,2,….,N–1. Evaluating, ( )t 1 u EU E ce E E b λ − = − Λ G G JJJJG Complementary (transient) solution Particular (steady-state) solution ( 1 1 where 1 1 j N T t t t t j ce c e c e c e c e λ λ λ λ − − = JJJJG j c − −1 = G ) 2 2 t N λ
Stability Analysis Stability Criterion We can think of the solution to the semi-discretized problem u=elce-EAE-b as a superposition of eigenmodes of the matrix operator a Each mode j contributes a(transient) time behaviour of the form e to the time-dependent part of the solution Since the transient solution must decay with time Real(2)≤0 or al This is the criterion for stability of the space discretization(of a parabolic PDE)keeping time continuous SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS We can think of the solution to the semi-discretized problem 15 Stability Analysis Stability Criterion ( )t 1 uE ce E E b λ − = Λ G JJJJG G This is the criterion for stability of the space discretization (of a parabolic PDE) keeping time continuous. Since the transient solution must decay with time, Real 0 ( ) for all j λ j ≤ Each mode contributes a (transient) time behaviour of the form to the time-dependent part of the solution. j t j e λ as a superposition of eigenmodes of the matrix operator A. −1 −