All Questions
Tagged with pde linear-algebra
24
questions
1
vote
0
answers
107
views
How to implement boundary conditions for the Thomas algorithm
For my variable $U(t,x)$, I have implemented the thomas algorithm with $U_j^i$:
$$ a(x)U_{j-1}^{i+1}+ b(x)U_j^{i+1} + c(x)U_{j+1}^{i+1} = d(x)U_j^{i} $$
Then $\textbf{A}$ is a tridiagonal vector with ...
0
votes
0
answers
59
views
Solving AU = F using linalg.cg results in 0 iterations
I am working on solving the following PDE: $$\left(\mu_{x}\frac{\partial^{2}u}{\partial x^{2}}+\mu_{y}\frac{\partial^{2}u}{\partial y^{2}}\right)=f(x,y) \tag 1$$
Which is then discretised:
$$- \mu_{x} ...
- 43
11
votes
1
answer
1k
views
Is using iterative methods to solve a linear system always superior to inversing the matrix?
I have a silly question. Is it always more computationally efficient to use iterative methods to solve for some matrix $A$, $Ax=b$, where $x$ and $b$ change but $A$ stays constant, compared to ...
- 131
4
votes
2
answers
166
views
Numerical estimation of eigenfunctions of Laplacian
Consider the Laplace equation,
$$
\nabla^2 f(r,\theta,\phi) = 0
$$
in spherical coordinates. We know that the solution to this equation can be derived analytically, and is given by,
$$
f(r,\theta,\phi)...
- 1,058
1
vote
1
answer
99
views
Norm of operator in finite element discretization of Heat equation
I am solving the heat equation discretized spatially via FEM and temporally via backward Euler.
I get the system
$$M \dot{u} = K u +f$$ where $u$ is a vector representing the solution at spatial ...
- 155
2
votes
1
answer
60
views
Discretization with non-constant matrix containg entries form unknown vector
Consider a system of PDEs
$$
\begin{cases}
u_t = \nabla \cdot (D(u)\nabla u) + \frac{c}{K_U+c}u-ku\\
c_t = d_c\Delta c -\frac{\nu_U c}{K_U + c}u
\end{cases}
$$
with some boundary conditions. Here, $D(...
- 226
0
votes
1
answer
182
views
Role of non-hermitian coefficient matrices in the discretization of self-adjoint operators
What is the role of non-symmetric coefficient matrices in the solution of partial differential equations with self-adjoint operators?
Particularly, I'm thinking about time-propagation of a linear ...
- 3,187
0
votes
1
answer
190
views
How to compute matrix representation of $\hat{y}\frac{\partial}{\partial x}$?
I have a 2-dimensional system which I would like to solve numerically (I'm using finite difference method right now), and its an eigenvalue problem. I have a term that looks like $H\psi(x,y) = [-\frac{...
- 101
2
votes
3
answers
313
views
Eigenvectors of Black-box matrix
$\DeclareMathOperator{\diag}{diag}$
Consider the generalized eigenproblem $A\mathbf{x}=\lambda B\mathbf{x}$. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet ...
- 63
4
votes
1
answer
455
views
Are the Finite Difference and Finite Volume Methods different after the application of the Gauss Divergence theorem on the FVM?
I have a rudimentary question about the differences between finite difference (FDM) and finite volume methods (FVM).
In FDM we concentrate on the nodes (points) in space while in FVM we concentrate ...
4
votes
2
answers
580
views
Well-posedness of a linear elasticity problem and Navier-Cauchy equation
I read a master thesis on a topic I'm interested too. This work concern the solution of the displacement equation of motion for a homogeneous, elastic, isotropic material:
$$\rho \ddot{\mathbf{u}} - \...
- 151
1
vote
2
answers
98
views
Performance metrics to compare initial-boundary value problem solutions
I am comparing the performance several finite difference methods of solving an initial-boundary value problem. There are several dimensions to this comparison:
Number of cells
Number of timesteps
...
- 469
5
votes
6
answers
489
views
Request algorithm recommomendation for 2D generalised Poisson solution
I need to examine the (static) electric field distribution around various electrode configurations in the presence of a dielectric, and have been using a finite difference approach to the 2D ...
- 63
10
votes
2
answers
1k
views
Is the maximum/minimum principle of the heat equation maintained by the Crank-Nicolson discretization?
I'm using the Crank-Nicolson finite difference scheme to solve a 1D heat equation. I'm wondering if the maximum/minimum principle of the heat equation (i.e. that the maximum/minimum occurs at the ...
- 105
5
votes
2
answers
4k
views
4th order Padé scheme formula derivation
I am trying to derive the formula of the 4th order Padé scheme that passes through the points $x_i$, $x_{i-1}$ and $x_{i+1}$
$$\Big(\frac{\partial\phi}{\partial x} \Big)_i = -\frac{1}{4}\Big(\frac{\...
- 1,029