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Quantitative Finance · Glossário

O que é Finite-difference method, method of lines, non-uniform grid?

Também chamado de: finite-difference method · method of lines · non-uniform grid

Definition 27.1 Quantitative Methods · Capítulo 27 — Finite-Difference Methods

A finite-difference method replaces the derivatives of a differential equation by difference quotients of the values on a grid of nodes. The method of lines discretises space first, turning the equation into a system of ordinary differential equations U˙=LhU\dot U = L_hU in time, then applies a time-stepping scheme. On a non-uniform grid the spacings hi−=xi−xi−1h_i^- = x_i - x_{i-1} and hi+=xi+1−xih_i^+ = x_{i+1} - x_i differ; the three-point formulas

uxx≈2ui−1h−(h−+h+)−2uih−h++2ui+1h+(h−+h+),u_{xx} \approx \frac{2u_{i-1}}{h^-(h^- + h^+)} - \frac{2u_i}{h^-h^+} + \frac{2u_{i+1}}{h^+(h^- + h^+)},
ux≈−h+ui−1h−(h−+h+)+(h+−h−)uih−h++h−ui+1h+(h−+h+)u_x \approx \frac{-h^+u_{i-1}}{h^-(h^- + h^+)} + \frac{(h^+ - h^-)u_i}{h^-h^+} + \frac{h^-u_{i+1}}{h^+(h^- + h^+)}

are exact on quadratics.

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