Local truncation error

Local truncation error is the discrepancy produced by a numerical method in one step when that step starts from the exact solution. It measures the method’s one-step approximation error, rather than accumulated error over many steps.

Numerical analysis: The study of algorithms for approximating mathematical problems and the errors those approximations produce. Local truncation error is a basic accuracy measure within this broader field.
Forward Euler method: A one-step method that advances an ordinary differential equation using its slope at the current point. Substituting the exact solution into its update leaves a defect proportional to the square of the step size.
Global truncation error: The difference between a numerical solution and the exact solution after multiple numerical steps. Unlike local truncation error, it includes error accumulated and propagated throughout the computation.
Adaptive step-size control: A strategy that changes numerical step sizes to meet a requested error tolerance efficiently. Solvers estimate local error to decide whether to accept a step or repeat it with a smaller interval.
Lax equivalence theorem: A theorem stating that a consistent finite difference scheme for a well-posed linear initial-value problem converges if and only if it is stable. It explains why vanishing local defect alone is insufficient for convergence without stability.
Initial value problem: A differential equation together with a specified solution value at one point. The exact solution of an initial value problem supplies the input used to define a one-step error.
Backward Euler method: An implicit one-step method that uses the derivative at the next time point. Its exact-solution substitution reveals the local defect while showing why implicitness does not guarantee higher order.
Round-off error: Error caused by finite-precision representation and arithmetic in a computer. Local truncation error comes from discretizing the equations, not from finite-precision arithmetic.
Embedded Runge–Kutta methods: Runge–Kutta schemes that compute two approximations of different orders using shared derivative evaluations. The difference between their approximations provides an inexpensive estimate of local error.
Error propagation: The evolution of perturbations or approximation errors through successive computational steps. Each step’s local truncation error joins errors already propagated from earlier steps.

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