Implicit multistep methods

Definition

Implicit multistep methods are numerical techniques used to solve ordinary differential equations (ODEs) by taking multiple steps forward in time and relying on the values of the solution at those steps, often including the unknown future value in their formulation. These methods can provide improved stability properties, especially for stiff equations, by allowing implicit equations that must be solved at each time step. The methods differ from explicit ones, as they require solving algebraic equations rather than just direct calculation.

5 Must Know Facts For Your Next Test

1. Implicit multistep methods can handle stiff problems more effectively than explicit methods, which can be crucial for certain applications in engineering and physics.
2. These methods involve the solution of a system of equations at each time step, often requiring iterative techniques like Newton's method.
3. The backward differentiation formula (BDF) is a popular type of implicit multistep method that is particularly well-suited for stiff ODEs.
4. Implicit methods generally require more computational resources per step due to the need to solve algebraic equations compared to explicit methods.
5. The order of accuracy for implicit multistep methods can be higher than that of explicit ones, making them advantageous for long-term integration problems.

Review Questions

• How do implicit multistep methods differ from explicit multistep methods in terms of stability and application?
  • Implicit multistep methods generally provide better stability properties, especially when dealing with stiff equations, where explicit methods may require prohibitively small time steps to maintain stability. While explicit methods compute the future values based solely on known previous values, implicit methods incorporate future unknowns and require solving systems of equations at each step. This difference makes implicit methods more suitable for certain applications where stiffness is a concern, such as in chemical kinetics or structural dynamics.
• Discuss the computational challenges associated with using implicit multistep methods compared to explicit methods.
  • Using implicit multistep methods involves solving algebraic equations at each time step, which can be computationally intensive and often requires iterative solvers like Newton's method. This contrasts with explicit methods, which simply calculate the next value based on known previous values without requiring complex computations. The increased computational cost of implicit methods is justified when dealing with stiff problems or when higher accuracy is required over long simulations.
• Evaluate the impact of using backward differentiation formulas (BDF) within the framework of implicit multistep methods on solving stiff ordinary differential equations.
  • Backward differentiation formulas (BDF) enhance the effectiveness of implicit multistep methods when tackling stiff ordinary differential equations by allowing for higher order accuracy while maintaining stability. BDF methods take advantage of previous time steps to predict future values while incorporating the stiffness characteristics directly into their formulation. This enables practitioners to simulate systems that exhibit rapid changes without compromising numerical stability, making BDF a preferred choice in fields like fluid dynamics and control theory where stiffness is prevalent.

Related terms

Stiff Equations: Equations that exhibit rapid changes in solutions, requiring smaller time steps for stable numerical solutions.

Adams-Bashforth Methods:

A family of explicit multistep methods that use previous values to estimate future values without involving future unknowns.

Runge-Kutta Methods:

A class of single-step methods used for solving ODEs that are known for their accuracy and flexibility.