Note on the Convergence of the Implicit Euler Method

• István Faragó
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)

Abstract

For the solution of the Cauchy problem for the first order ODE, the most popular, simplest and widely used method are the Euler methods. The two basic variants of the Euler methods are the explicit Euler methods (EEM) and the implicit Euler method (IEM). These methods are well-known and they are introduced almost in any arbitrary textbook of the numerical analysis, and their consistency is given. However, in the investigation of these methods there is a difference in concerning the convergence: for the EEM it is done almost everywhere but for the IEM usually it is missed. (E.g., [1, 2, 6-9].)The stability (and hence, the convergence) property of the IEM is usually shown as a consequence of some more general theory. Typically, from the theory for the implicit Runge-Kutta methods, which requires knowledge of several basic notions in numerical analysis of ODE theory, and the proofs are rather complicated. In this communication we will present an easy and elementary prove for the convergence of the IEM for the scalar ODE problem. This proof is direct and it is available for the non-specialists, too.

Keywords

Numerical solution of ODE implicit and explicit Euler method Runge-Kutta methods finite difference method

References

1. 1.
Ascher, U.M., Petzold, L.R.: Computer methods for ordinary differential equations and differential-algebraic equations. SIAM, Philadelphia (1998)
2. 2.
Bachvalov, N. S.: Numerical methods, Nauka, Moscow (1975) (in Russian)Google Scholar
3. 3.
Dahlquist, G.: Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 4, 33–53 (1956)
4. 4.
Collected Works of Euler, L. vol. 11 (1913), 12 (1914) pp. 549–553 (1984)Google Scholar
5. 5.
Isaacson, E., Keller, H.B.: Analysis of numerical methods. Wiley, New York (1966)
6. 6.
LeVeque, R.: Finite difference mehods for ordinary and partial differential equations. SIAM, Philadelphia (2007)Google Scholar
7. 7.
Lambert, J.D.: Numerical methods for ordinary differential systems: The initial value problem. John Wiley and Sons, Chicester (1991)
8. 8.
Molchanov, I.N.: Computer methods for solving applied problems. Differential equations. Naukova Dumka, Kiew (1988) (in Russian)Google Scholar
9. 9.
Samarskij, A.A., Gulij, A.V.: Numerical methods, Nauka, Moscow (1989) (in Russian)Google Scholar
10. 10.
Suli, E.: Numerical solution of ordinary differential equations, Oxford (2010)Google Scholar