3.1 Adams-Bashforth Methods

Adams-Bashforth methods are explicit linear multistep techniques for solving differential equations. They work by integrating an interpolating polynomial that passes through previous solution points, offering a balance between accuracy and computational efficiency.

These methods build on single-step approaches, using information from multiple past steps to predict future values. While they can provide higher accuracy, they also have unique stability considerations that impact their use in different problem types.

Adams-Bashforth Methods Derivation

Interpolating Polynomials and Integration

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• Adams-Bashforth methods are explicit linear multistep methods derived from integrating an interpolating polynomial that passes through the solution at previous time steps
• The derivation involves approximating the integral of the derivative function using an interpolating polynomial (Lagrange or Newton form) and the method of undetermined coefficients
  • The interpolating polynomial is constructed to match the solution values at the required past time steps
  • The coefficients of the polynomial are determined by solving a system of linear equations arising from the interpolation conditions
• The order of the Adams-Bashforth method is determined by the degree of the interpolating polynomial used in the derivation
  • Higher-order methods use higher-degree polynomials, requiring more past solution values
  • Common orders for Adams-Bashforth methods are 2, 3, 4, and 5

Coefficients and Past Values

• The coefficients of the Adams-Bashforth methods are obtained by solving a system of linear equations that arise from the interpolation conditions and the integration of the polynomial
  • The coefficients depend on the order of the method and the choice of interpolation points
  • The coefficients are typically pre-computed and stored for efficient implementation
• The Adams-Bashforth methods require the solution values at multiple previous time steps, with higher-order methods requiring more past values
  • For example, a 4th-order Adams-Bashforth method requires the solution at the current time step and the three previous time steps
  • The past solution values are used to construct the interpolating polynomial and evaluate the derivative function

Implementing Adams-Bashforth Methods

Explicit Scheme and Time Stepping

• Adams-Bashforth methods are implemented as explicit schemes, where the solution at the current time step is computed using the solution values at previous time steps
  • The solution at the current time step is obtained by a weighted sum of the derivative function evaluated at the past time steps
  • The weights are determined by the coefficients of the Adams-Bashforth method
• The implementation requires storing the solution values at the necessary previous time steps and evaluating the derivative function at those points
  • The past solution values are typically stored in an array or buffer that is updated at each time step
  • The derivative function is evaluated at the past time steps using the stored solution values

Accuracy and Stability Considerations

• The choice of the time step size and the order of the method affects the accuracy and stability of the numerical solution
  • Smaller time steps generally lead to more accurate solutions but increase the computational cost
  • Higher-order methods provide better accuracy for smooth solutions but may be less stable for stiff problems
• Higher-order Adams-Bashforth methods provide better accuracy but may require smaller time steps to maintain stability
  • The stability region of Adams-Bashforth methods decreases with increasing order
  • The time step size should be chosen carefully to ensure stability, especially for stiff problems
• The initial steps of the Adams-Bashforth method may require a different starting procedure, such as using a single-step method (Runge-Kutta methods), to obtain the required past solution values
  • The starting procedure should be of the same order of accuracy as the Adams-Bashforth method to maintain overall accuracy
  • Once the required past solution values are obtained, the Adams-Bashforth method can be applied for the subsequent time steps

Adams-Bashforth Error Analysis

Local Truncation Error

• The local truncation error of an Adams-Bashforth method is the error introduced in a single step of the method, assuming exact solution values are used at the previous time steps
  • It measures the accuracy of the method in approximating the true solution over one step
  • The local truncation error is derived by comparing the numerical solution with the Taylor series expansion of the true solution
• The order of accuracy of an Adams-Bashforth method is determined by the leading term in the local truncation error expression
  • The leading term is proportional to a power of the time step size, denoted as $O([h](https://www.fiveableKeyTerm:h)^{p+1})$, where $p$ is the order of the method
  • Higher-order methods have a higher power of $h$ in the leading term, indicating faster convergence as the time step size decreases

Global Error and Numerical Verification

• The global error of an Adams-Bashforth method, which accumulates over multiple steps, is typically one order lower than the local truncation error
  • The global error at a specific time is the difference between the numerical solution and the true solution at that time
  • The global error is influenced by the accumulation of local truncation errors and the stability of the method
• The order of accuracy can be verified numerically by comparing the errors obtained with different time step sizes and observing the rate at which the error decreases
  • By halving the time step size and comparing the errors, the observed order of accuracy can be estimated
  • The observed order of accuracy should approach the theoretical order as the time step size decreases, confirming the accuracy of the implementation

Adams-Bashforth vs Single-Step Stability

Stability Regions and Stiffness

• Adams-Bashforth methods have different stability properties compared to single-step methods like Runge-Kutta methods
  • The stability of a numerical method refers to its ability to produce bounded solutions for stable problems
  • Adams-Bashforth methods have smaller stability regions compared to implicit methods, which can limit the choice of time step sizes for stiff problems
• The stability of Adams-Bashforth methods depends on the order of the method and the properties of the problem being solved
  • Higher-order Adams-Bashforth methods generally have more restrictive stability requirements compared to lower-order methods
  • Stiff problems, characterized by widely varying time scales or rapidly decaying components, can pose challenges for explicit methods like Adams-Bashforth

Stability Analysis Techniques

• The stability of Adams-Bashforth methods can be analyzed using techniques such as the root condition, which examines the roots of the characteristic polynomial associated with the method
  • The root condition requires that all roots of the characteristic polynomial lie within the stability region of the method
  • Violation of the root condition indicates potential instability and may require a reduction in the time step size
• In some cases, Adams-Bashforth methods may require smaller time steps to maintain stability compared to single-step methods, especially for stiff problems or problems with rapidly varying solutions
  • The stability limitations of Adams-Bashforth methods can be mitigated by using smaller time steps or by switching to implicit methods (Adams-Moulton methods) for stiff problems
  • Adaptive time stepping strategies can be employed to automatically adjust the time step size based on stability and accuracy considerations