Backward Euler method

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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. Consider the ordinary differential equation d y d t = f ( t , y ) {\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} t}}=f(t,y)} with initial value y ( t 0 ) = y 0 . {\displaystyle y(t_{0})=y_{0}.} Here the function f {\displaystyle f} and the initial data t 0 {\displaystyle t_{0}} and y 0 {\displaystyle y_{0}} are known; the function y {\displaystyle y} depends on the real variable t {\displaystyle t} and is unknown. A numerical method produces a sequence y 0 , y 1 , y 2 , … {\displaystyle y_{0},y_{1},y_{2},\ldots } such that y k {\displaystyle y_{k}} approximates y ( t 0 + k h ) {\displaystyle y(t_{0}+kh)} , where h {\displaystyle 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 ) . {\displaystyle y_{k+1}=y_{k}+hf(t_{k+1},y_{k+1}).} This differs from the (forward) Euler method in that the forward method uses f ( t k , y k ) {\displaystyle f(t_{k},y_{k})} in place of f ( t k + 1 , y k + 1 ) {\displaystyle f(t_{k+1},y_{k+1})} . The backward Euler method is an implicit method: the new approximation y k + 1 {\displaystyle 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 {\displaystyle y_{k+1}} . For non-stiff problems, this can be done with fixed-point iteration: y k + 1 [ 0 ] = y k , y k + 1 [ i + 1 ] = y k + h f ( t k + 1 , y k + 1 [ i ] ) . {\displaystyle y_{k+1}^{[0]}=y_{k},\quad y_{k+1}^{[i+1]}=y_{k}+hf(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 {\displaystyle y_{k+1}} . Alternatively, one can use (some modification of) the Newton–Raphson method to solve the algebraic equation. For a linear differential equation, f ( y , t ) = p ( t ) y ( t ) + q ( t ) {\displaystyle f(y,t)=p(t)y(t)+q(t)} , the update step can be written explicitly as y k + 1 = y k + h q ( t k + 1 ) 1 − h p ( t k + 1 ) . {\displaystyle y_{k+1}={\frac {y_{k}+hq(t_{k+1})}{1-hp(t_{k+1})}}.} Integrating the differential equation d y d t = f ( t , y ) {\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} t}}=f(t,y)} from t n {\displaystyle t_{n}} to t n + 1 = t n + h {\displaystyle t_{n+1}=t_{n}+h} yields y ( t n + 1 ) − y ( t n ) = ∫ t n t n + 1 f ( t , y ( t ) ) d t . {\displaystyle 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 ) ≈ h f ( t n + 1 , y ( t n + 1 ) ) . {\displaystyle y(t_{n+1})-y(t_{n})\approx hf(t_{n+1},y(t_{n+1})).} Finally, use that y n {\displaystyle y_{n}} is supposed to approximate y ( t n ) {\displaystyle y(t_{n})} and the formula for the backward Euler method follows.