Extrapolation Let IR denote the exact solution of an equation, denotes the grid size of a numerical approximation scheme, and () the

Extrapolation Let η∗ ∈ IR denote the exact solution of an equation, Δ denotes the grid size of a numerical approximation scheme, and η(Δ) the approximating solution. Further assume an error model η(Δ) − η∗ = c Δq, with

c, q ∈ IR. q is the order of the approximation scheme. Suppose that for two grid sizes

Δ1, Δ2 = 1 2 Δ1 approximations η1 := η(Δ1), η2 := η(Δ2) are calculated.

a) For the case of a known η∗ (or η∗ approximated with very high accuracy) establish a formula for the order q out of η∗, η1, η2.

b) For a known order q show that η∗ = 1 2q − 1 (2qη2 − η1). In general, the error model holds only approximately. Hence this formula for η∗ is only an approximation to the exact η∗ (“extrapolation”).