16.920J/SMA 5212 Numerical Methods for PDEs STABILITY ANALYSIS Some Important Characteristics Deduced A few features worth considering 1. Stability analysis of time discretization scheme can be carried out for all the different modes 1 2. If the stability criterion for the time discretization scheme is valid for all modes, then the overall solution is stable(since it is a linear combination of all the modes) 3. When there is more than one root o, then one of them is the principal root which represents an approximation to the physical behaviour The principal root is recognized by the fact that it tends towards one as Ah0, i.e. lim o(h)=1.(The other roots are spurious, which affect the stability but not the accuracy of the scheme Slide 33 STABILITY ANALYSIS Some Important Characteristics Deduced 4. By comparing the power series solution of the principal root to e one can determine the order of accuracy of the time discretization cheme. In this example of leapfrog time discretization, a=h+(1+h2)=+1+1{(h2)+2,2h h2 and compared to hi is identical up to the second order of h/. Hence, the above scheme is said to be second-order accurate Slide 3416.920J/SMA 5212 Numerical Methods for PDEs 21 STABILITY ANALYSIS Some Important Characteristics Deduced A few features worth considering: 1. Stability analysis of time discretization scheme can be carried out for all the different modes . 2. If the stability criterion for the time discretization scheme is λ j valid for all modes, then the overall solution is stable (since it is a linear combination of all the modes). 3. When there is more than one root , then one of them is the principal root which represents σ ( ) 0 an approximation to the physical behaviour. The principal root is recognized by the fact that it tends towards one as 0, i.e. lim 1. (The other roots are spurious, which affect the stability h h h λ λ σ λ → → = but not the accuracy of the scheme.) Slide 33 STABILITY ANALYSIS Some Important Characteristics Deduced 1 4. By comparing the power series solution of the principal root to , one can determine the order of accuracy of the time discretization scheme. In this example of leapfrog time discretization, 1 h e h λ σ = λ + ( ) ( ) 1 2 2 2 2 4 4 2 2 2 1 2 2 1 1 . 1 2 2 1 . 2 2! 1 ... 2 and compared to 1 ... 2! is identical up to the second order of . Hence, the above scheme is said to be second-order accurate. h h h h h h h h e h h λ λ λ λ λ λ σ λ λ λ λ − + = + + + = + + + = + + + Slide 34