Trunction errors in numerical integration

Numerical integration

In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...

 are of two kinds:

• local truncation errors – the error caused by one iteration, and
• global truncation errors – the cumulative error cause by many iterations.

Definitions

Suppose we have a continuous differential equation

and we wish to compute an approximation of the true solution at discrete time steps . For simplicity, assume the time steps are equally spaced

Suppose we compute the sequence with

Note: A is called an increment function, and can be interpreted as an estimate of the slope of y(t).

Local truncation error

The local truncation error is the error that our increment function, A, causes during a given iteration, assuming perfect knowledge of the true solution at the previous iteration.

More formally, the local truncation error, , at step is defined by:

Global truncation error

The global truncation error is the accumulation of the local truncation error over all of the iterations, assuming perfect knowledge of the true solution at the initial time step.

More formally, the global truncation error, , after steps is defined by:

Relationship between local and global truncation errors

Sometimes it is possible to calculate an upper bound on the global truncation error, if we already know the local truncation error. This requires our increment function be sufficiently well-behaved.

See also

• Numerical integration
  Numerical integration
  In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
  
  
• Numerical ordinary differential equations
  Numerical ordinary differential equations
  Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...
  
  
• Truncation error
  Truncation error
  Truncation error or local truncation error is error made by numerical algorithms that arises from taking finite number of steps in computation...
  
  

External links

• Notes on truncation errors and Runge-Kutta methods

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