# Tag Info

Accepted

### Motivation behind Collocation Method

I would not insist on demanding a geometrical meaning from Galerkin methods in general. There is a connection, but it becomes less meaningful as you extend it further and further. (In a sense, it ...
• 12.3k
Accepted

### Don't we care about the numerical diffusion in the diffusion term?

We have the following problem: $$\frac{\partial u}{\partial t}+v\color{red}{\frac{\partial u}{\partial x}}-\nu\color{blue}{\frac{\partial^2u}{\partial x^2}}=0 \tag{*}$$ The function $u$ may represent ...
• 1,638
Accepted

### How do I integrate a function defined over an arbitrary area?

Instead of directly integrating over the area, it is often more convenient to use the divergence theorem to replace the area integral with an integral over the boundary edges. The divergence theorem ...
• 6,064

### Why do we have to resort to Higher order schemes for solving the 1-D advection equation/ continuity equation?

There is a difference between the requirements for a hyperbolic pde like $$u_t + a u_x = 0$$ and for a purely parabolic pde like $$u_t = u_{xx}$$ Suppose the solutions are smooth and you ...
• 3,028

### What is numerical damping in the context of time-dependent FEM solvers?

It is quite straightforward to demonstrate this for implicit Euler (aka backward Euler) on a scalar example. Consider the initial value problem $$\dot{y}(t) = i \alpha y(t), \ y(0) = 1$$ with solution ...
• 1,273
Accepted

• 8,672

### How to isolate and test time discretization order of accuracy

If you fix the spatial discretization, then you are solving a fixed system of ODEs. The exact solution of that system is not the same as the exact solution of the PDE, of course, but you can test ...
• 16.5k

### Stability Analysis

This is a first-order PDE but you are trying to impose Dirichlet BC on the whole boundary. This is not well-posed in general but maybe your data is such that you expect it to have a unique solution. ...
• 3,028
Accepted

• 3,127
Accepted

### Discretizing the viscous component in 1 - D Navier stokes compressive flow

Consider a FV method as an approximation of the integral conservation law. Starting from the one-dimensional, scalar conservation equation $$u_t + f(u,\nabla u)_x =0,$$ ...
• 1,287

### Can I use Q0 finite elements when there are gradients involved?

There are two ways for what you are trying to do: If you approximate the gradient by finite differences involving the values at the cell midpoints, then your use of Q0 elements will lead to typical ...
• 55.5k

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