What is convection-dominated pde problems?

Question

1. Can you explain for me what is convection-dominated problems? Definition and examples if possible.

2. Why don't we can apply standard discretization methods (finite difference, finite element, finite volume methods) for convection-dominated equation?

3. What is the importance of global Péclet number in this problem?

Answer 1

Suppose we are modeling a quantity $u$, say the concentration of a chemical, driven by a flow in some fluid in a region $\Omega$ with no source (meaning we are not adding more chemical into the fluid after starting the timer). Then the convection-diffusion pde to describe the phenomenon is: $$ \frac{\partial u}{\partial t} = \nabla \cdot (D \nabla u - \vec{b}u ).\tag{1} $$ This comes from linear hyperbolic conservation law (in the differential form): $$ \frac{\partial u}{\partial t} = -\nabla \cdot \vec{F}, $$ where $\vec{F}$ is the flux vector, and $$ \int_{\partial \Omega}\vec{F}\cdot n\,dS $$ measure the amount of chemical flows out from the domain of interest $\Omega$, and $\vec{F} = -D \nabla u + \vec{b}u$. $D$ is the diffusion constant, it describes how the quantity diffuses, $\vec{b}$ is a flow field, it may carry $u$ around. If we assume $D$ is a constant and $\vec{b}$ is a divergence free flow (incompressible), then (1) is: $$ \frac{\partial u}{\partial t} = \underbrace{D\Delta u}_{\text{Diffusion term}} - \underbrace{\vec{b}\cdot \nabla u}_{\text{Convection term}}. \tag{2} $$ Convection-dominance just means, in the convection-diffusion equation (2): $$D\ll \|\vec{b}\|.$$ The diffusion term is very small relative to the convection term.

Intuitively, diffusion means "smooth", while convection could possibly contain non-smoothness due to its derivation from an integral conservation law. For example, if we make the diffusion term $D\sim 0$ almost gone, the solution may looks like the following in a 2D problem:

where the solution's discontinuity "flows" with the field $\vec{b}$ on the right. If the diffusion constant is huge, i.e., not convection-dominated, then the steep slope will be smoothened as the quantity being transported along with the flow $\vec{b}$.

---

As I said in the comments, traditional numerical methods relies on the smoothness of the solution. Solution like above can't be solved by tradition numerical schemes like you listed. That's why people using Discontinuous Galerkin finite element to numerically solve this type of problems, while we don't impose the continuity conditions of the numerical solutions, in the meantime the conservation can be guaranteed numerically.

For the last question, please just read 2.1.4 here on page 33.