Survey of Spectral Approximations

Abstract

Now that we know how to approximate functions, integrals and derivatives with high order orthogonal functions, we move to our ultimate goal and develop methods to approximate the solutions of partial differential equations (PDEs). In this book, we will concentrate on the spectral approximation of three basic equations of mathematical physics, namely the potential equation ∇ 2 ϕ = s, (4.1) the advection-diffusion equation ϕ t + q · ∇ϕ = ν∇ 2 ϕ, (4.2) where q is some velocity field, and the wave equation ϕ tt − c 2 ∇ 2 ϕ = 0. (4.3) From the advection-diffusion equation we can immediately reduce to the scalar ad-vection problem (ν = 0) or the diffusion problem/heat equation (q = 0). The form of the equations (4.1)-(4.3), the form in which the equations are usually written, is called the strong form. The strong form of the equations may require the solutions to be more smooth than we want. For instance the temperature, ϕ, in a thin insulated rod of length, L, with variable thermal diffusivity and zero temperature specified at the ends is described by the initial-boundary value problem ⎧ ⎪ ⎨ ⎪ ⎩ ϕ t = νϕ xx + ν x ϕ x , x ∈ (0, L) , t > 0, ϕ(0, t) = ϕ(L, t) = 0, ϕ(x, 0) = ϕ 0 (x). (4.4) Its classical solution is the one that must be at least twice differentiable in space. The diffusivity, ν > 0, must also be differentiable. We know physically, however, that at the joint between two materials with different thermal conductivities, ν will not be differentiable and a slope discontinuity will appear in the solution to make the heat flux, f = νϕ x , continuous. The piecewise smooth solution, which makes perfect physical sense, is not a classical solution to the heat equation. To include solutions with weaker smoothness constraints, we need to rewrite the equations. To allow a larger class of solutions, we rewrite the PDE in a weak form. For example, if we write the right hand side of the PDE in (4.4) in the form (νϕ x) x and