10.2 Adaptive step-size algorithms

Adaptive step-size algorithms are game-changers for solving differential equations. They adjust the step size on the fly, balancing accuracy and efficiency. This means smaller steps when things get wild and bigger steps when it's smooth sailing.

These algorithms use clever tricks like comparing different order approximations to estimate errors. They then tweak the step size to keep errors in check. It's like having a smart autopilot for your numerical methods!

Error Estimation and Step Size Control

Variable Step Size and Error Estimation

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• Variable step size allows the step size to change during the integration process based on the estimated error
• Smaller step sizes are used when the solution changes rapidly or has high curvature to maintain accuracy
• Larger step sizes are used when the solution changes slowly to improve efficiency and reduce computational cost
• Error estimation involves comparing two different order approximations of the solution at each step
  • The difference between these approximations provides an estimate of the local truncation error
  • Common methods for error estimation include local extrapolation and embedded Runge-Kutta methods

Step Size Control and Local Extrapolation

• Step size control adjusts the step size based on the estimated error to maintain a desired level of accuracy
  • If the estimated error is larger than a specified tolerance, the step size is reduced
  • If the estimated error is smaller than the tolerance, the step size can be increased to improve efficiency
• Local extrapolation estimates the error by comparing two approximations of different orders
  • For example, comparing a 4th-order Runge-Kutta method with a 5th-order Runge-Kutta method
  • The difference between these approximations provides an estimate of the local truncation error

Richardson Extrapolation

• Richardson extrapolation is a technique for improving the accuracy of numerical approximations
• It involves combining two approximations of different step sizes to obtain a higher-order approximation
• The extrapolated solution has a higher order of accuracy than the original approximations
• Richardson extrapolation can be used to estimate the error and control the step size in adaptive step-size algorithms
  • By comparing the extrapolated solution with one of the original approximations, an estimate of the error can be obtained
  • This error estimate can then be used to adjust the step size to maintain a desired level of accuracy

Embedded Runge-Kutta Methods

Overview of Embedded Runge-Kutta Methods

• Embedded Runge-Kutta methods are a class of adaptive step-size algorithms that use two Runge-Kutta methods of different orders
• The two methods share the same stages (function evaluations) but have different coefficients
• The difference between the two approximations provides an estimate of the local truncation error
• Embedded Runge-Kutta methods are efficient because they reuse the function evaluations for both the solution and error estimation
• Examples of embedded Runge-Kutta methods include Fehlberg methods, Cash-Karp methods, and Dormand-Prince methods

Dormand-Prince Method

• The Dormand-Prince method is a widely used embedded Runge-Kutta method
• It uses a 5th-order Runge-Kutta method for the solution and a 4th-order Runge-Kutta method for error estimation
• The Dormand-Prince method has seven stages (function evaluations) shared between the two methods
• The coefficients of the Dormand-Prince method are chosen to minimize the error of the 5th-order solution
• The difference between the 5th-order and 4th-order solutions provides an estimate of the local truncation error
• The Dormand-Prince method is the default method used in many software packages for solving ordinary differential equations (MATLAB's

  ode45

  , SciPy's

  integrate.solve_ivp

  )