1D Discretization Sample Clauses

1D Discretization. Let us consider the 1D discretization. If we take the model problem: ∂ u ∂2u ∂t − µ∂x2 = f (3.8) We replace the variable u representing the solution to the problem, with uh a piecewise polynomial. Let us introduce a mesh. h is a representative dimension of the size of the elements. We can consider uh as a sum of Lagrange piecewise polynomials. uh = Σ Uj(t)φj(x) (3.9) Here φj(x) are the Lagrange piecewise polynomials, meaning φj = 1 at node xj and 0 on all other nodes, and is piecewise linear around the node xj. Now we will look at the weak formulation: ∂t φi + = ∂x fφi (3.10) ∫ b ∂uh ∫ b ∂ uh ∂ φi ∫ b a a ∂x a and thus Σ ∫ b dUj(t) j a ∫ b ∂ φj ∂ φi ∫ b a a ( dt φjφi + Uj ∂x = ∂x fφi (3.11) Now we can define the matrix M with the entries ∫ b Mij = φjφi (3.12) a K, the matrix with the entries ∫ b ∂ φj ∂ φi a ∂x Kij = (3.13) ∂x and vector b with entries bi = ∫ fφi (3.14) a Then we are left with the ordinary differential equation system: dU M + KU = b. (3.15) dt Under the theory of FDM we now perform a time discretization with a θ method. As stated above the FDM approximates the derivative and solves considering the following general equation: lim ∆x→0 yn+1 − yn ∆x = f (y n+1 , xn+1 ) (3.16) The theta method considers the following: 