16.920J/SMA 5212 Numerical Methods for PDEs E△E Complementary Particular(steady-state) (transient )solution solution where (cy)=[ The stability analysis of the space discretization, keeping time continuous, is based on the eigenvalue structure of a The exact solution of the system ofequations is determined by the eigenvalues and eigenvectors ofa STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs We can think of the solution to the semi-discretized problem u=Elce-EA-E-lb as a superposition of eigenmodes of the matrix operator A Each mode j contributes a( transient )time behaviour of the form d to the time-dependent part of the solution Since the transient solution must decay with time Real()≤0 for all j This is the criterion for stability of the space discretization(of a parabolic PDE) keeping time continuous Slide 516.920J/SMA 5212 Numerical Methods for PDEs 11 Evaluating, ( ) t 1 1 u EU E ce E E b λ − − = = − Λ ✂ ✂ ￾✁￾✁￾￾✂ ✂ ( ) 1 2 1 w 1 2 1 here j N T t t t t t j N ce c e c e c e c e λ λ λ λ λ − − ✄ ☎ = ✆ ✝ ✞✁✞✁✞✟✞✠ The stability analysis of the space discretization, keeping time continuous, is based on the eigenvalue structure of A. The exact solution of the system of equations is determined by the eigenvalues and eigenvectors of A. Slide 14 STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs We can think of the solution to the semi-discretized problem as a superposition of eigenmodes of the matrix operator A. Each mode contributes a (transient) time behaviour of the form to the time-dependent part of the solution. j t j e λ Since the transient solution must decay with time, Real ( ) 0 λ j ≤ for all j This is the criterion for stability of the space discretization (of a parabolic PDE) keeping time continuous. Slide 15 Complementary (transient) solution Particular (steady-state) solution ( ) t 1 1 u E ce E E b λ − − = − Λ ☛ ✡✁✡✁✡✟✡☛ ☛