3.2 Adams-Moulton Methods

Adams-Moulton methods are a family of implicit linear multistep techniques for solving differential equations. They work by integrating the equation over a step and using an interpolating polynomial to approximate the integral, offering improved stability and accuracy over explicit methods.

These methods are particularly useful for stiff problems, allowing larger step sizes while maintaining stability. However, they require solving a nonlinear equation at each step, which can be computationally expensive. The trade-off between stability and computational cost is a key consideration when choosing numerical methods.

Adams-Moulton Methods

Interpolating Polynomial Construction

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• Adams-Moulton methods are a family of implicit linear multistep methods for solving ordinary differential equations
• These methods are derived by integrating the differential equation over a step and approximating the integral using an interpolating polynomial
• The interpolating polynomial is constructed using the values of the solution and its derivative at the current and previous steps
  • The values used depend on the specific Adams-Moulton method (e.g., 2-step or 3-step method)
• The order of the method is determined by the degree of the interpolating polynomial used
  • Higher-order methods use higher-degree polynomials, resulting in greater accuracy but increased computational complexity
• The implicit nature of Adams-Moulton methods requires solving a nonlinear equation at each step, typically using an iterative method such as Newton's method or fixed-point iteration

Method Properties and Characteristics

• Adams-Moulton methods are implicit, meaning the solution at the current step depends on the solution at the same step
  • This is in contrast to explicit methods, where the solution at the current step depends only on the solution at previous steps
• The coefficients of the Adams-Moulton methods are derived by imposing consistency conditions on the interpolating polynomial
  • These conditions ensure that the method accurately approximates the integral of the differential equation over the step
• Adams-Moulton methods have good stability properties, allowing for larger step sizes compared to explicit methods
  • This is particularly advantageous for stiff differential equations, where the solution has components with widely varying time scales
• The order of accuracy of Adams-Moulton methods increases with the degree of the interpolating polynomial
  • Higher-order methods have a higher order of accuracy, meaning the error decreases more rapidly as the step size is reduced

Implicit Adams-Moulton Applications

Solving Initial Value Problems

• Adams-Moulton methods are well-suited for solving initial value problems, where the solution and its derivative are known at an initial point
• The general form of an Adams-Moulton method is a linear combination of the solution and its derivative at the current and previous steps
  • The coefficients of the linear combination are determined by the order of the method and the step size
• Implementing Adams-Moulton methods requires a starting procedure, such as Runge-Kutta methods, to obtain the initial values of the solution and its derivative
  • The starting procedure provides the necessary values to begin the multistep method
• The implicit equation at each step is solved using an iterative method, with the solution from the explicit Adams-Bashforth method serving as an initial guess
  • The Adams-Bashforth method provides a good initial approximation for the iterative solver, reducing the number of iterations required

Computational Considerations

• Adams-Moulton methods require the solution of a nonlinear equation at each step, which can be computationally expensive
  • The cost of solving the nonlinear equation depends on the chosen iterative method and the desired accuracy
• The implicit nature of Adams-Moulton methods allows for larger step sizes compared to explicit methods, reducing the total number of steps required
  • This can offset the increased computational cost per step, especially for stiff problems or problems with long integration intervals
• Adams-Moulton methods can be combined with adaptive step size control to optimize the trade-off between accuracy and efficiency
  • Adaptive step size control adjusts the step size based on error estimates, ensuring that the desired accuracy is maintained while minimizing the number of steps

Error and Accuracy of Adams-Moulton Methods

Local Truncation Error

• The local truncation error of an Adams-Moulton method is the error introduced in a single step, assuming the previous values are exact
• The local truncation error can be expressed as the difference between the true solution and the numerical solution at the end of the step
  • This error arises from the approximation of the integral using the interpolating polynomial
• The local truncation error depends on the order of the method and the step size
  • Higher-order methods have a lower local truncation error for a given step size
• The local truncation error can be estimated using Taylor series expansions or by comparing the numerical solution with a higher-order reference solution

Order of Accuracy

• The order of accuracy of an Adams-Moulton method is the exponent of the step size in the leading term of the local truncation error
• The order of accuracy determines the rate at which the local truncation error decreases as the step size is reduced
  • Doubling the step size for a method of order p will increase the local truncation error by a factor of 2^(p+1)
• Adams-Moulton methods of order p have a local truncation error proportional to the (p+1)th power of the step size
  • For example, the 2-step Adams-Moulton method (trapezoidal rule) has a local truncation error proportional to h^3, where h is the step size
• The global error, which is the accumulation of local truncation errors over all steps, is also influenced by the order of accuracy
  • The global error of an Adams-Moulton method of order p is proportional to h^p, assuming the step size is sufficiently small

Stability of Adams-Moulton vs Explicit Methods

Stability Properties

• Stability is a crucial property of numerical methods for solving differential equations, as it determines the behavior of the numerical solution in the presence of perturbations
• Adams-Moulton methods are implicit methods, which generally have better stability properties than explicit methods
  • Implicit methods allow for larger step sizes while maintaining stability, particularly for stiff problems
• The stability regions of Adams-Moulton methods are larger than those of explicit Adams-Bashforth methods, allowing for larger step sizes while maintaining stability
  • The stability region is the set of complex values for which the method produces bounded solutions when applied to the test equation y' = λy
• The improved stability of Adams-Moulton methods is particularly advantageous for stiff differential equations, where the solution has components with widely varying time scales
  • Stiff problems often require very small step sizes for explicit methods to maintain stability, making them computationally inefficient

Computational Trade-offs

• The implicit nature of Adams-Moulton methods requires more computational effort per step compared to explicit methods, but this is often offset by the ability to use larger step sizes while maintaining accuracy and stability
  • The larger stability regions of Adams-Moulton methods allow for larger step sizes, reducing the total number of steps required
• The choice between explicit and implicit methods depends on the specific problem and the desired balance between accuracy, stability, and computational efficiency
  • For non-stiff problems or problems with relaxed accuracy requirements, explicit methods may be more efficient due to their lower computational cost per step
• In practice, a combination of explicit and implicit methods can be used, such as predictor-corrector schemes
  • The explicit method (predictor) provides an initial approximation, which is then refined by the implicit method (corrector), combining the efficiency of explicit methods with the stability of implicit methods