All Questions
Tagged with finite-difference linear-algebra
28
questions
1
vote
0
answers
107
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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 ...
3
votes
1
answer
391
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Generalized eigenvalue problem for large, potentially ill-conditioned systems
Say that I have a generalized eigenvalue problem of the form $$Ax=\lambda Bx.$$ Using MATLAB, some naive ways that one may solve this is by either
directly inverting $B$ then applying the ...
- 43
11
votes
1
answer
1k
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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
1
vote
1
answer
262
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constructing a symmetric matrix for finite difference
I come across the following operator in a paper
$\mathcal{I}\psi = \psi_{xxxx} + (r~\psi_x)_x$,
where $\psi=\psi(x)$ and $r=r(x)$. Periodic boundary condition is employed. It claims that the operator $...
- 217
3
votes
1
answer
73
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Requesting for Finite Difference Methods reference in Portuguese or English
Crossposted on Mathematics SE
I have been assigned a group project for an introductory Linear Algebra subject on Finite Difference Methods and sparse matrices. Our professor advised we use Gilbert ...
- 33
2
votes
0
answers
76
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How to solve this boundary value problem which has more unknown than equation on MATLAB
I need your helps about solving the problem below with MATLAB. I am trying to solve 2D Stress Wave Propagation problem by using FDTD(Finite difference time domain) method on the cylindrical coord. I ...
- 21
0
votes
0
answers
253
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How to make a directed graph symmetric?
Say I have a directed graph given as an adjacency matrix $A$ in CSR format represented by the arrays ia (row indexes) and ja (...
- 617
3
votes
1
answer
153
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Numerical calculation of the Berry connection
I'm doing some numerical calculations involving Hermitian matrices, and derivatives of the eigenvectors.
Essentially, I have an n x n, Hermitian matrix H(x), which is dependent on some continuous ...
1
vote
1
answer
99
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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
3
votes
2
answers
1k
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Solving Ax = b with sparse A and sparse b
Let's suppose I'm numerically solving the Poisson equation for a delta function source:
$$ \nabla^2 f(x) = \delta(x-x') $$
I can represent the Laplacian $\nabla^2$ using the finite difference method ...
- 203
6
votes
1
answer
230
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Does mean removal increase accuracy of numerical differentiation?
I wish to compute the derivative of a vector through numerical differentiation. Let's say, we use a standard 2nd order central difference scheme, to arrive at a differentiation matrix, and apply it on ...
2
votes
2
answers
572
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Precision loss in Matrix-Vector product when applying Finite-Difference scheme
I am applying a 6th order Finite-Difference differentiation scheme as seen in http://www.scholarpedia.org/article/Method_of_lines/example_implementation/dss006
There seems to be severe numerical/...
0
votes
1
answer
190
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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
4
votes
1
answer
1k
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Thomas algorithm for 3D finite difference
For 1D finite difference, the resulting linear system is tri-diagonal and can be solved in O(n) using the Thomas algorithm.
I am trying to solve a finite ...
- 615
1
vote
2
answers
194
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How to determine the truncation error with products and quotients
If I have an equation given by
$$\displaystyle Y = \frac{a^2}{d^2}\frac{(1-c^2\frac{c}{a})}{(1-b^2)}$$
and I expand $a,b,c,d$ in a Taylor series, where $a$ is truncated at the $A^{th}$ order, $b$ is ...
- 183