Discretization and Rounding Errors ODEs and Beyond Alan

„Discretization and Rounding Errors: ODE´s and Beyond“ Alan Krautstengl Institute of Finance and Administration Czech Republic Institute of Computer Science of the Czech Academy of Sciences Czech Republic

Errors the Mathematical Modeling and Scientific Computing

Real World Problem Ø Mathematical model (Integro-differencial equations) § Infinite – dimensional problem § Existence and uniqueness of solution Ø Numerical model (Galerkin, Ritz, FEM, FD) § Finite – dimensional approximation § Convergence to analytic solution Ø Linear algebraic model (System of lin. equations) § Forming of the linear system § Matrix computation

Error Analysis 1. Mathematical model (Errors of the model) 2. Numerical model (Discretization errors) 3. Linear algebraic model (Computational errors)

Total Error TOTAL ERROR = MODEL ERROR + DISCRETIZATION ERROR + COMPUTATION ERROR Ø Not independent § Importance of understanding how they may influence each other Ø Good solution process carries fundamental properties (e. g. positive definiteness) throughout the embedding scheme

ODE´s Ø Mutual relationship between discretization and computational errors is well-understood Ø interesting influence

Ingredients Ø Algebra Ø Analysis Similar holds for general one-step and multi-step method Ø Higher dimesions? § General result non-existent § Marginal treatment in leading texts (e. g. Jim Demmel – „Applied Numerical Linear Algebra“)

Algebraic and Analytical Treatment Method of lines Finite differences Finite elements Ø Analysis get complicated Ø Intuition and special cases suggest similarities

Other Problems Ø Other problems § Choice of norm § Algorithm dependence of rounding errors Acknowledgement Author wishes to acknowledge Zdeněk Strakoš for fruitful discussions and mathematical insight contributing to this presentation