15. Adams–Moulton Method
The chapter discusses the Adams-Bashforth method, an explicit multistep technique for the numerical solution of ordinary differential equations (ODEs). It highlights the advantages of using such methods for accurate long-term integrations while addressing their limitations regarding stability and initialization. The chapter concludes with insights on the accuracy, error analysis, and applications of the Adams-Bashforth method across various fields.
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What we have learnt
- Adams-Bashforth method is an explicit multistep approach for solving ODEs.
- This method is more efficient than single-step methods for long-time integrations.
- Understanding and applying this technique requires careful consideration of stability and initial conditions.
Key Concepts
- -- Multistep Methods
- Techniques that utilize multiple previous points to compute the next value of a solution for algorithms dealing with ODEs.
- -- AdamsBashforth Method
- An explicit predictor-based multistep method that estimates future values of a function using its previous derivative values.
- -- Local Truncation Error (LTE)
- The error made in one step of a numerical method, indicating how the numerical solution compares to the exact solution at that step.
- -- Global Error
- The total error of a numerical solution over all steps, summing up the local errors.
- -- Explicit Methods
- Numerical methods where the next value is calculated directly from known previous values.
- -- Implicit Methods
- Numerical methods where the next value depends on itself, leading to potential complexities in solving.
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