{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

28-Higher Order Methods

28-Higher Order Methods - Runge-Kutta methods Multistep...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Runge-Kutta methods Multistep methods Predictor-corrector methods Higher-Order Methods Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Higher-Order Methods MATH 2070U 1 / 19 Runge-Kutta methods Multistep methods Predictor-corrector methods Higher-Order Methods 1 Runge-Kutta methods 2 Multistep methods 3 Predictor-corrector methods c D. Aruliah (UOIT) Higher-Order Methods MATH 2070U 2 / 19 Runge-Kutta methods Multistep methods Predictor-corrector methods Reminder: Numerical solution of IVPs IVP y = f ( t , y ( t )) , t I , y ( t 0 ) = y 0 . Mesh t 0 < t 1 < · · · < t N h with step-size h t n = t 0 + nh ( n = 0: N h ) Numerical solution: { un } N h n = 0 such that u n y ( t n ) Time-stepping: using recurrence to compute u n + 1 from u n ( n = 0: N h - 1); u 0 = y 0 c D. Aruliah (UOIT) Higher-Order Methods MATH 2070U 3 / 19 Runge-Kutta methods Multistep methods Predictor-corrector methods Reminder: Some one-step methods Prototypical one-step methods for time-stepping: u n + 1 = u n + h f n forward Euler (explicit) u n + 1 = u n + h f n + 1 backward Euler (implicit) u n + 1 = u n + h 2 ( f n + f n + 1 ) Crank-Nicolson (implicit) Errors: local e n = y ( t n ) - u n , global max 0 n N h | e n | Accuracy of discretisations measured by convergence of errors Forward Euler: global error = O ( h ) Backward Euler: global error = O ( h ) Crank-Nicolson: global error = O ( h 2 ) c D. Aruliah (UOIT) Higher-Order Methods MATH 2070U 4 / 19
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Runge-Kutta methods Multistep methods Predictor-corrector methods Construction of one-step methods Set up integral t n + 1 t n f ( τ , y ( τ )) d τ = t n + 1 t n y ( τ ) d τ = y ( t n + 1 ) - y ( t n )
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.
  • Spring '10
  • aruliahdhavidhe
  • Numerical differential equations, Heun's method, Numerical ordinary differential equations, Multistep Methods, D. Aruliah

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern