Linear multistep method

Definition

A linear multistep method is a numerical technique used to solve ordinary differential equations (ODEs) by approximating solutions using multiple previous points. These methods employ a linear combination of past values of the solution and its derivatives to produce an estimate for the next value, making them efficient for solving initial value problems. By utilizing information from multiple steps, these methods can achieve higher accuracy than single-step methods.

5 Must Know Facts For Your Next Test

1. Linear multistep methods can be categorized into explicit and implicit methods, depending on how they utilize previous points in calculations.
2. The order of a linear multistep method refers to the highest derivative used in the approximation, impacting both accuracy and stability.
3. These methods can suffer from issues such as numerical instability if not implemented carefully, especially when used for stiff equations.
4. The convergence of linear multistep methods relies heavily on the consistency and stability properties of the method chosen.
5. Common examples of linear multistep methods include the Adams-Moulton and Adams-Bashforth methods, each with different characteristics and applications.

Review Questions

• How do linear multistep methods differ from single-step methods in terms of their approach to solving differential equations?
  • Linear multistep methods differ from single-step methods by using multiple previous solution values and their derivatives to compute the next step. This allows them to take into account more information from earlier steps, which can lead to higher accuracy. In contrast, single-step methods, like the Runge-Kutta method, only use information from the most recent point to calculate the next value.
• Discuss the significance of stability and convergence in relation to linear multistep methods when applied to stiff differential equations.
  • Stability and convergence are crucial when applying linear multistep methods to stiff differential equations because these equations exhibit rapid changes that can cause numerical instability if not handled properly. Implicit methods, such as Backward Differentiation Formulas (BDF), are often preferred in this context due to their inherent stability. Understanding how the properties of a specific linear multistep method influence stability is essential for obtaining reliable numerical solutions.
• Evaluate how the choice between explicit and implicit linear multistep methods affects computational efficiency and accuracy in practical applications.
  • The choice between explicit and implicit linear multistep methods significantly influences computational efficiency and accuracy. Explicit methods, like Adams-Bashforth, are generally simpler and faster because they do not require solving a system of equations at each step. However, they may become unstable for stiff problems. In contrast, implicit methods offer improved stability, especially for stiff equations, but involve more complex calculations that can increase computational time. Balancing these trade-offs is essential when selecting an appropriate method for specific problems.

Related terms

Adams-Bashforth Method: A specific type of linear multistep method that is explicit and uses past solution values to estimate future values.

Backward Differentiation Formula (BDF): An implicit linear multistep method that uses previous points to provide stability for stiff differential equations.

Runge-Kutta Method: A family of single-step methods for solving ODEs that can achieve higher accuracy but rely on evaluating the derivative multiple times per step.