Exponential scheme

• Write the steady 1D advection-diffusion equation, given below, in terms of the total flux $F$, $$u\frac{d\phi}{dx} =\Gamma\frac{d^2\phi}{dx^2}.$$
• Write the finite volume form of this equation in terms of the flux $F$.
• This equation has an exact solution for Dirichlet boundary conditions.
  • T/F: we can apply the exact solution between grid cells to evaluate the face values for $\phi$ and $d\phi/dx$.
  • T/F: these face values from the exact solution are an improvement over simple interpolations and can be used even when the overall ODE or PDE is more complex than the 1D steady advection-diffusion equation.
• T/F: exponentials are expensive, but can be replaced with piecewise linear functions or polynomials.
• Sketch the form of the exact solution of $\phi(x)$ for varying $Pe$.
  • Explain physically in terms of advection and diffusion why the gradient is sharp at the edges for high $Pe$.
• Incorporating exact solutions for simpler problems into numerical methods for complex problems is a powerful approach. Consider other areas where such thinking may be possible.