Predictor–corrector method

In numerical analysis, predictor–corrector methods belong to a class of algorithms designed to integrate ordinary differential equations to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps: # The initial, "prediction" step, starts from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate ("anticipate") this function's value at a subsequent, new point. # The next, "corrector" step refines the initial approximation by using the predicted value of the function and another method to interpolate that unknown function's value at the same subsequent point.

In numerical analysis, predictor–corrector methods belong to a class of algorithms designed to integrate ordinary differential equations to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps: The initial, "prediction" step, starts from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate ("anticipate") this function's value at a subsequent, new point. The next, "corrector" step refines the initial approximation by using the predicted value of the function and another method to interpolate that unknown function's value at the same subsequent point. y' = f(t,y), \quad y(t_0) = y_0, and denote the step size by h . First, the predictor step: starting from the current value y_i , calculate an initial guess value \tilde{y}_{i+1} via the Euler method, \tilde{y}_{i+1} = y_i + h f(t_i,y_i). Next, the corrector step: improve the initial guess using trapezoidal rule, y_{i+1} = y_i + \tfrac12 h \bigl( f(t_i, y_i) + f(t_{i+1},\tilde{y}_{i+1}) \bigr). That value is used as the next step. \tilde{y}_{i+1} &= y_i + h f(t_i,y_i), \\ y_{i+1} &= y_i + \tfrac12 h \bigl( f(t_i, y_i) + f(t_{i+1},\tilde{y}_{i+1}) \bigr). \end{align} It is also possible to evaluate the function f only once per step by using the method in Predict–Evaluate–Correct (PEC) mode: \begin{align} \tilde{y}_{i+1} &= y_i + h f(t_i,\tilde{y}_i), \\ y_{i+1} &= y_i + \tfrac12 h \bigl( f(t_i, \tilde{y}_i) + f(t_{i+1},\tilde{y}_{i+1}) \bigr). \end{align} Additionally, the corrector step can be repeated in the hope that this achieves an even better approximation to the true solution. If the corrector method is run twice, this yields the PECECE mode: \begin{align} \tilde{y}_{i+1} &= y_i + h f(t_i,y_i), \\ \hat{y}_{i+1} &= y_i + \tfrac12 h \bigl( f(t_i, y_i) + f(t_{i+1},\tilde{y}_{i+1}) \bigr), \\ y_{i+1} &= y_i + \tfrac12 h \bigl( f(t_i, y_i) + f(t_{i+1},\hat{y}_{i+1}) \bigr). \end{align} The PECEC mode has one fewer function evaluation than PECECE mode. More generally, if the corrector is run k times, the method is in P(EC) k or P(EC) k E mode. If the corrector method is iterated until it converges, this could be called PE(CE) ∞ .

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