This work presents procedures for estimating the error of numerical solutions of multi-dimensional problems. It is considered that: the numerical error is caused only by truncation errors; error estimations are based on the Richardson extrapolation; and numerical approximations are one-dimensional over uniform grids in each dimension. Two cases are analyzed: when grids are simultaneously refined in all four dimensions (x,y,z,t); and when grid refinement in each dimension is separate from the remaining ones. Examples of uses are presented for problems involving heat transfer and fluid mechanics, which are solved by the finite difference and finite volume methods. It was found that, for the situation in which the apparent order of the estimated error is a monotone convergent one, two values of estimated error can be calculated, which bound the true error.
Discretization error; truncation error; CFD; numerical error; fluid flows
