Adams-Bashforth Methods

Numerical Analysis · Axiom Academy

LESSON Adams-Bashforth and Adams-Moulton Methods Multi-step methods for solving ODEs: leveraging past solution values for efficient, high-order numerical integration Adams-Bashforth (AB) methods are explicit multi-step methods. They compute the next value using only previously known function evaluations, making them straightforward to implement. The general idea: integrate the ODE from x_n to x_ n+1 using a polynomial that interpolates past derivative values f(x_k, y_k). Adams-Moulton (AM) methods are implicit multi-step methods. They include f(x_ n+1 , y_ n+1 ) in the formula, requiring an iterative solution but achieving higher accuracy for the same number of steps. Adams-Moulton 2-step (AM2) - Trapezoidal: 3. Predictor-Corrector Approach The predictor-corrector method combines the best of both worlds: use Adams-Bashforth to predict an initial estimate, then use Adams-Moulton to correct it. Predict with AB4: y*_ n+1 = y_n + h/24[55f_n - 59f_ n-1 + 37f_ n-2 - 9f_ n-3 ] Evaluate : f*_ n+1 = f(x_ n+1 , y*_ n+1 ) Correct with AM4: y_ n+1 = y_n + h/24[9f*_ n+1 + 19f_n - 5f_ n-1 + f_ n-2 ] Optionally : iterate correction until convergence 4. Coefficients and Error Terms The coefficients for Adams methods are derived from integrating Lagrange interpolating polynomials. Here are the commonly used formulas with their local truncation errors: 5. Comparison with Runge-Kutta Methods

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