Completed notes of the course Numerical Methods

Pagina 1

1 NOTES OF NUMERICAL METHODS Prof. P. Zunino Part [1] – FINITE DIFFERENCE DISCRETIZATIONS 0. REVIEW OF ODEs Numerical problems In order to solve with a calculator a mathematical problem we have to discretize it, because a calculator doesn’t work with continuous functions but with numerical problem, so we have to find some approximations of the derivaives which transform the analytical problem into a numerical one. 𝑁𝑢𝑚𝑒𝑟𝑖𝑐𝑎 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑖𝑧𝑎𝑡𝑖𝑜𝑛: 𝑦(𝑥!,𝑡")≅𝑢!" Approximation of first order derivatives In order to discretize ODE and PDE problems we use three different finite difference approximation of first order derivatives: 𝐹𝑜𝑟𝑒𝑤𝑎𝑟𝑑 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛: 𝐷#𝑦(𝑡)=𝑦(𝑡+ℎ)−𝑦(𝑡)ℎ 𝐵𝑎𝑐𝑘𝑤𝑎𝑟𝑑 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛: 𝐷$𝑦(𝑡)=𝑦(𝑡)−𝑦(𝑡−ℎ)ℎ 𝐶𝑒𝑛𝑡𝑒𝑟𝑒𝑑 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛: 𝐷%𝑦(𝑡)=𝑦(𝑡+ℎ)−𝑦(𝑡−ℎ)2ℎ Where h is the distance between two nodes in space Order of accuracy The order of accuracy p of an approximation is the speed with which the error committed by the approximation goes to zero: 𝑦&(𝑡)=𝐷𝑦(𝑡)+𝑂(ℎ') 𝐹𝑜𝑟𝑒𝑤𝑎𝑟𝑑 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛: 𝐸=𝑂(ℎ() 𝐵𝑎𝑐𝑘𝑤𝑎𝑟𝑑 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛: 𝐸=𝑂(ℎ() 𝐶𝑒𝑛𝑡𝑒𝑟𝑒𝑑 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛: 𝐸=𝑂(ℎ)) ´ The order of accuracy can be estimated using a truncated Taylor expansion Approximation of second order derivatives Also the second order derivative can be discretized with a finite difference approximation: 𝑆𝑒𝑐𝑜𝑛𝑑 𝑜𝑟𝑑𝑒𝑟 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛: 𝐷)𝑦(𝑡)=𝑦(𝑡+ℎ)−2𝑦(𝑡)+𝑦(𝑡−ℎ)ℎ) Where h is the distance between two nodes in space Order of accuracy The order of accuracy p of an approximation is the speed with which the error committed by the approximation goes to zero: 𝑦&′(𝑡)=𝐷)𝑦(𝑡)+𝑂(ℎ') 𝑆𝑒𝑐𝑜𝑛𝑑 𝑜𝑟𝑑𝑒𝑟 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛: 𝐸=𝑂(ℎ)) 2 Euler schemes for ODEs In order to solve numerically simple ODE problems we can use the Euler methods, which consist in using the three finite difference approximation of first order derivatives to discretize the…