For linear multistep methods, an additional concept called zero-stability is needed to explain the relation between local and global truncation errors.
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Truncation errors are committed when an iterative method is terminated or a mathematical procedure is approximated, and the approximate solution differs from the exact solution.
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A slightly different formulation for the local truncation error can be obtained by using the Lagrange form for the remainder term in Taylor's theorem.
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Thus, for extremely small values of the step size, the truncation error will be small but the effect of rounding error may be big.
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A finite difference scheme is compact in the sense that the discretised formula comprises at most nine point truncation error terms in the finite difference equation.
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In explicit filtering, an LES filter is applied to the discretized Navier Stokes equations, providing a well-defined filter shape and reducing the truncation error.
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If the solution y has a bounded second derivative and f is Lipschitz continuous in its second argument, then the global truncation error ( GTE ) is bounded by
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The global truncation error is the error at a fixed time t, after however many steps the methods needs to take to reach that time from the initial time.
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Often, truncation error also includes discretization error, which is the error that arises from taking a finite number of steps in a computation to approximate an infinite process.
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Occasionally, round-off error ( the consequence of using finite precision floating point numbers on computers ) is also called truncation error, especially if the number is rounded by truncation.
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