1. SCIENTIFIC DISCOVERY EXPERIMENT THEORY SCIENTIFIC COMPUTING 1

2. SOLUTION COMPUTER MODEL (ALGORITHM) MATHEMATICAL MODEL EXPERIMENTAL DATA DISCRETE MODEL 2

3. SOLUTION COMPUTER MODEL (ALGORITHM) MATHEMATICAL MODEL EXPERIMENTAL DATA DISCRETE MODEL EXPERIMENTAL ERRORS UNCERTAINTIES MODELING ERRORS DICRETIZATION ERRORS ROUND-OFF ERRORS UNCERTAINTIES +MODELING ERRORS +DISCRETIZATION ERRORS +ROUND-OFF ERRORS 3

4. SOLUTION COMPUTER MODEL (ALGORITHM) MATHEMATICAL MODEL EXPERIMENTAL DATA DISCRETE MODEL EXPERIMENTAL ERRORS UNCERTAINTIES Uncertainty Quantification MODELING ERRORS Validation DICRETIZATION ERRORS Verification ROUND-OFF ERRORS Numerical Stability UNCERTAINTIES +MODELING ERRORS +DISCRETIZATION ERRORS +ROUND-OFF ERRORS Accuracy 4

5. VALIDATION: Are we solving the correct equation? --Reduction of modeling errors VERIFICATION: Are we solving the equation correctly? --Reduction of discretization errors UNCERTAINTY QUANTIFICATION: Characterization of uncertainties in problem ---Variability of input and/or model parameters ERROR CONTROL PROCESSES 5

6. Example Pendulum L : Length of pendulum m : Mass of pendulum g : Gravitational acceleration a : Initial angle θ’’(t) + m g/L sin(θ(t))=0, t >0, Θ(0)= a Θ’(0)=0 θ Mathematical Model θ‘’ + m g /L θ =0, t >0 θ(0)=a θ’(0)=0 Discrete Model, RK Methods 6