Tagged Questions
7
votes
2answers
93 views
Is the maximum/minimum principle of the heat equation maintained by the Crank-Nicolson discretization?
I'm using the Crank-Nicolson finite difference scheme to solve a 1D heat equation. I'm wondering if the maximum/minimum principle of the heat equation (i.e. that the maximum/minimum occurs at the ...
15
votes
2answers
197 views
Why do equi-spaced points behave badly?
Description of experiment:
In Lagrange interpolation, the exact equation is sampled at $N$ points (polynomial order $N - 1$) and it is interpolated at 101 points. Here $N$ is varied from 2 to 64. ...
1
vote
1answer
89 views
rate of convergence for the second order accurate method on two dimensional grid
I am solving $u_t=u_{xx}+u_{yy}$ with a solver which is locally $O(dx^2+dy^2+dt^2)$. I use the following norm for the error between the numerical vector solution and the analytical solution ...
8
votes
2answers
268 views
Alternatives to von neumann stability analysis for finite difference methods
I'm working on solving the coupled one-dimensional poroelasticity equations (biot's model), given as:
$$-(\lambda+ 2\mu) \frac{\partial^2 u}{\partial x^2} + \frac{\partial p}{\partial x} = 0$$
...
2
votes
2answers
167 views
I don't understand how to correctly calculate truncation error
I am looking at the finite difference methods to solve simple $u_t=a(x,t)u_{xx}$.
There are explicit, implicit, Crank Nicolson.
The latter is said to be more accurate since the local truncation ...
7
votes
4answers
167 views
How can I derive a bound on the spurious oscillations in the numerical solution of the 1D advection equation?
Suppose I had the following periodic 1D advection problem:
$\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$ in $\Omega=[0,1]$
$u(0,t)=u(1,t)$
$u(x,0)=g(x)$
where $g(x)$ has a ...
12
votes
3answers
578 views
How can I numerically differentiate an unevenly sampled function?
Standard finite difference formulas are usable to numerically compute a derivative under the expectation that you have function values $f(x_k)$ at evenly spaced points, so that $h \equiv x_{k+1} - ...
