16.920J/SMA 5212 Numerical Methods for PDEs STABILITY ANALYSIS Use of Modal (scalar) Equation It may be noted that since the solution iyTs expressed as a contribution from all the modes of the initial solution which have propagated or(and) diffused with the eigenvalue A. and a contribution from the source term b. all the properties of the time integration (and their stability properties) can be analysed separately for each mode with he scalar equation dU dt Slide 16 STABILITY ANALYSIS Use of Modal ( scalar) Equation The spatial operator A is replaced by an eigenvalue 1, and the above modal equation will serve as the basic equation of the stability of a time-integ (yet to be introduced) as a function of the eigenvalues a of the space-discretization operators This analysis provides a general technique for the determination of time integration methods which lead to stable algorithms for a given space discretization Slide 7 EXAMPLE 1 Continuous Time Operator Consider a set of coupled ODEs (2 equations only) a1l1+a12l2 dt 141 +all du lide 1816.920J/SMA 5212 Numerical Methods for PDEs 12 STABILITY ANALYSIS Use of Modal (Scalar) Equation It may be noted that since the solution is expressed as a contribution from all the modes of the initial solution, which have propagated or (and) diffused with the eigenvalue , and a contribution fr j u λ ￾ om the source term , all the properties of the time integration (and their stability properties) can be analysed separately for each mode with the scalar equation j b Slide 16 STABILITY ANALYSIS Use of Modal (Scalar) Equation The spatial operator A is replaced by an eigenvalue λ, and the above modal equation will serve as the basic equation for analysis of the stability of a time-integration scheme (yet to be introduced) as a function of the eigenvalues λ of the space-discretization operators. This analysis provides a general technique for the determination of time integration methods which lead to stable algorithms for a given space discretization. Slide 17 EXAMPLE 1 Continuous Time Operator Consider a set of coupled ODEs (2 equations only): 1 11 1 12 2 2 21 1 22 2 du a u a u dt du a u a u dt = + = + 1 11 12 2 21 22 Let , u a a du u A Au u a a dt ✁✄✂ ✁ ✂ = = ☎ = ✆ ✝ ✆ ✝ ✞✄✟ ✞ ✟ ✠ ✠ ✠ Slide 18 j dU U F dt λ ✡ ☛ = + ☞ ✌ ✍ ✎