Single step methods

1. Single step methods For first order ODE Single step method Local truncation error -- consistency Convergence Stability --- constraint on time step k!! Convergence rate or order of accuracy

2. Convergence An example Forward Euler method Exact solution Numerical solution The error

3. Convergence For single-step method Thm: Suppose the order of accuracy of the above single step method is p>0 and the incremental function satisfying the Lipschitz condition for y, i.e. In addition, suppose be exact. Then we have the global truncation error Proof: See details in class of as an exercise

4. Applications For forward Euler method Convergence if f(t,y) is Lipschitz continuous for y!! RK2 & RK4 Convergence if f(t,y) is Lipschitz continuous for y!!

5. Stability In computation, we have round-off error!!! Def: For a numerical method, if there is a perturbation at step , then the perturbation at all steps after is not larger than . Then the method is called as stable. Analyze the stability of numerical methods Use the model problem Apply the method to this problem Find the amplification factor

6. Stability For forward Euler method For model problem The method The amplification factor Stability condition & stability region

7. Stability For backward Euler method For model problem The method The amplification factor No stability condition for k (unconditionally stable!!) & stability region

8. Stability Trapezoidal method -- unconditionally stable RK4 For implicit methods Unconditionally stable!! For explicit methods Stability condition I-Stable methods RK3, RK4, implicit methods

9. Numerical example Conclusion: There is no stability condition for Backward Euler method There is stability condition for Forward Euler and RK4 Choice of time step Accuracy For explicit methods Must satisfying the stability condition!!

10. Numerical example The problem

11. Time-splitting (split-step) method The problem Integrate over time integral Formal exact solution

12. Time-splitting method First order splitting method Step 1: Solve Step 2: Solve Approximation to the original problem Local truncation error (see details in class)

13. Time-splitting method Second order splitting method (Strang splitting) Step 1: Solve Step 2: Solve Step 3: Solve Local truncation error (see details in class)

14. Time-splitting method Comments When A & B are commute, the splitting methods are exact!! They are very useful in solving PDEs For each subproblem, we can solve them either analytically or numerically For dispersive problems, we can design high order splitting, e.g. 4th order or 6th order splitting methods For dissipative problems, usually, we can only use second order splitting method.

15. Integration factor (IF) method The problem Moving the linear term to the left hand side Multiplying at both sides

16. Integration factor (IF) method Integrating over time interval Multiplying both sides by Apply numerical quadrature to the last term An example:

17. Multi-step methods The problem An m-step multistep method: is one whose difference equation for finding the approximation at the time step can be represented as Constants to be determined

18. Multi-step methods Explicit method: Implicit method: Ways to determine the constants Taylor expansion for local truncation error Function interpolation via polynomial

19. Multi-step methods Adams-Bashforth (AB) method � explicit, (r+1)-step Choose r+1 interpolation nodes Construct a polynomial based on the above nodes Numerical methods Order of accuracy & Stability Explicit

20. Multi-step methods Two-step Adams-Bashforth (AB2) method: r=1 2 interpolation points Interpolation polynomial Numerical method Order of accuracy: 2; Explicit; Stability region (see details in class)

21. Multi-step methods Four-step Adams-Bashforth (AB4) method: r=3 4 interpolation points Interpolation polynomial Numerical method Order of accuracy: 4; Explicit; Stability region (exercise)

22. Multi-step methods Adams-Moulton (AM) method � implicit, (r+1)-step Choose r+2 interpolation nodes Construct a polynomial based on the above nodes Numerical methods Order of accuracy & Unconditionally stable Implicit

23. Multi-step methods Two-step Adams-Moulton (AM2) method: r=1 3 interpolation points Interpolation polynomial Numerical method Order of accuracy: 3; Implicit & Unconditionally stable!!

24. Multi-step methods Four-step Adams-Moulton (AM4) method: r=3 5 interpolation points Interpolation polynomial Numerical method Order of accuracy: 5; Implicit; Unconditionally stable!!

25. Multi-step methods Adams-Bashforth methods Explicit Stability condition Adams-Moulton methods Implicit Unconditionally stable Same number of points with one order high accuracy Predictor-corrector methods Combine the advantages of both AB and AM methods Use AB methods to predict an intermediate value US AM methods to correct the prediction Usually, we use the same steps AB and AM methods

26. Multi-step methods Adams two-step predictor-corrector method AB2 for prediction AM2 for correction Properties Explicit, second order accurate, better stability than AB2!!