Backward Euler method

In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an implicit method. The backward Euler method has error of order one in time.

In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an implicit method. The backward Euler method has error of order one in time. \frac{\mathrm{d} y}{\mathrm{d} t} = f(t,y) with initial value y(t_0) = y_0. Here the function f and the initial data t_0 and y_0 are known; the function y depends on the real variable t and is unknown. A numerical method produces a sequence y_0, y_1, y_2, \ldots such that y_k approximates y(t_0+kh) , where h is called the step size. The backward Euler method computes the approximations using y_{k+1} = y_k + h f(t_{k+1}, y_{k+1}). This differs from the (forward) Euler method in that the forward method uses f(t_k, y_k) in place of f(t_{k+1}, y_{k+1}) . The backward Euler method is an implicit method: the new approximation y_{k+1} appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown y_{k+1} . For non-stiff problems, this can be done with fixed-point iteration: y_{k+1}^{[0]} = y_k, \quad y_{k+1}^{[i+1]} = y_k + h f(t_{k+1}, y_{k+1}^{[i]}). If this sequence converges (within a given tolerance), then the method takes its limit as the new approximation y_{k+1} . Alternatively, one can use (some modification of) the Newton–Raphson method to solve the algebraic equation. For a linear differential equation, the update step can be written explicitly as y_{k+1} = \frac{y_k + h q(t_{k+1})}{1 - h p(t_{k+1})}. Integrating the differential equation \frac{\mathrm{d} y}{\mathrm{d} t} = f(t,y) from t_n to t_{n+1} = t_n + h yields y(t_{n+1}) - y(t_n) = \int_{t_n}^{t_{n+1}} f(t, y(t)) \,\mathrm{d}t. Now approximate the integral on the right by the right-hand rectangle method (with one rectangle): y(t_{n+1}) - y(t_n) \approx h f(t_{n+1}, y(t_{n+1})). Finally, use that y_n is supposed to approximate y(t_n) and the formula for the backward Euler method follows. The same reasoning leads to the (standard) Euler method if the left-hand rectangle rule is used instead of the right-hand one. \begin{array}{c|c} 1 & 1 \\ \hline & 1 \\ \end{array} The method can also be seen as a linear multistep method with one step. It is the first method of the family of Adams–Moulton methods, and also of the family of backward differentiation formulas. ==See also== Crank–Nicolson method

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