Reference no: EM132252884

Questions -

Q1. Triangular Domains

We consider for sufficiently smooth data f and g_1,2 the 2D Poisson equation Δu = f on the triangular domain Ω = {(x, y) ∈ R²₊ : x + y < 1} with mixed boundary conditions

u(x, 0) = g₁(x), 0 < x < 1;

u(0, y) = g₂(y), 0 < y < 1;

∂_nu(x, 1 - x) = 0, 0 < x < 1.

Discretise this equation using a Neumann boundary condition analogy to the Shortley-Weller approximation and determine the consistency order of the resulting method. Formulate a matrix-free algorithm for the solution of this problem, based on the Jacobi or Gauβ-Seidel method.

Q2. Rothe's Method

The idea behind Rothe's Method for the heat equation

∂u/∂t = Δu, (x, y) ∈ Ω, t > 0, u|_∂Ω = g, u(0, x, y) = u₀(x, y)

is to consider this equation an ODE on a suitable function space and to first integrate in time. Using for this the implict Euler method one obtains the sequence of PDEs with solutions u^k+1(x, y)

u^k+1 = u^k + τΔu^k+1, (x, y) ∈ Ω, u^k+1|_∂Ω = g, u⁰(x, y) = u(0, x, y)

Discretise these PDEs using the 5-point Finite Difference formula and answer the following questions: What is the convergence order of this approach as a solution of the Heat Equation? For which choice of τ is the method linearly stable? Compare this method with the implict method from class.

Q3. 1D convection-diffusion-reaction problems

Let b > 0 be a constant and f : R⁺₀ → R⁺₀ be a continuous function that satisfies a Lipschitz condition. We consider on the interval (0, 1) the two-point boundary value problem

u'' - bu' = f(u), 0 < x < 1, u(0) = 1, u'(1) = 0

(a) Discretise this problem using the second order central finite differences for the 2nd derivative and for the first derivative on the LHS each of (i) the central difference formula, (ii) the forward difference formula, (iii) the backward difference formula.

(b) For each of the approximations (i)-(iii) in (a) determine whether (or: for which choice of h) a discrete maximum principle is satisfied.

Attachment:- Assignment Files.rar