Use of Modal (Scalar) Stability Analysis Equation It may be noted that since the solution u 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 from the source term b. all the properties of the time integration(and their stability properties) can be analysed separately for each mode with the scalar equation du nu +F SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS 16 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 G 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 j dU U F dt λ = +
Use of Modal (Scalar) Stability Analysis Equation The spatial operator A is replaced by an eigenvalue A, 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 n 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 SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS 17 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