Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.
3
votes
1answer
48 views
Conservative finite-difference expression for the advection equation
Following on from the earlier question I am trying to derive a finite-difference scheme for the advection equation which is conservative. It was suggested that for advection equation with variable ...
2
votes
0answers
29 views
Explicit 4th order space wave equation not stable implementation?
The explicit 4th order discretization for the 2D scalar wave equation is given by:
\begin{eqnarray}
U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} }{\Delta s} \right) ^2 \left( \sum_{a=-N}^N w_a ...
1
vote
0answers
54 views
What methods exist to solve for the fluid flow past a cylinder using finite differences on a Cartesian grid?
I'm interested in finite-difference approaches to the incompressible Navier-Stokes equations that can handle complex geometry without the use of an unstructured mesh or a non-Cartesian grid. To be ...
3
votes
1answer
57 views
Trouble implementing Neumann boundary conditions because the ghost points cannot be eliminated
Neumann boundary conditions are implemented by introducing ghost points outside the domain and then using the boundary conditions to eliminate the ghost points. For example, see this question.
I ...
2
votes
0answers
51 views
Von Newman stability analysis for 2D acoustic wave equation explicit
Von Newman stability analysis for acoustic wave equation explicit centered differences: 2nd order time and space (N 2)'th order:
\begin{eqnarray}
U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} ...
7
votes
2answers
90 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 ...
3
votes
1answer
67 views
Are there any numerical advantages in solving symmetric matrix compared to matrices without symmetry?
I'm applying finite-difference method to a system of 3 coupled equations. Two of the equations are not coupled, however the third equation couples to both the other two. I noticed that by changing the ...
3
votes
1answer
38 views
Closed form for singular values of 2D Laplacian?
Does anyone know where to find an analytic form for the singular values of the finite-difference approximation to the 2D Laplacian, expressed in matrix form for a square grid? This would be for the ...
5
votes
1answer
161 views
Implementing Explicit formulation of 1D wave equation in Matlab
So the theory is straightforward. We have:
$$\frac{\partial^2U}{\partial t^2}=c^2 \frac{\partial^2U}{\partial x^2}$$
discretizing it gives:
$$\frac{U(i+1,j)- 2U(i,j) + U(i-1,j)}{(\Delta t)^2} = c^2 ...
1
vote
0answers
77 views
Block Cyclic Reduction implementation help, matlab
I'm sorry if this question is too open ended, or not appropriate for the forum. If so I'll remove the post and see if I can refine my problem, but at the moment I'm totally stumped.
I was hoping ...
1
vote
1answer
61 views
Von Neumann Stability Analysis
I came across the following task recently:
Use the von-Neumann stability analysis to investigate the stability of the discrete form of $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial ...
6
votes
1answer
130 views
How to approximate the condition number of a large matrix?
How do I approximate the condition number of a large matrix $G$, if $G$ is a combination of Fourier transforms $F$ (non-uniform or uniform), finite differences $R$, and diagonal matrices $S$?
The ...
1
vote
1answer
76 views
4th order Padé scheme formula derivation
I am trying to derive the formula of the 4th order Padé scheme that passes through the points $x_i$, $x_{i-1}$ and $x_{i+1}$
$$\Big(\frac{\partial\phi}{\partial x} \Big)_i = ...
1
vote
1answer
69 views
Singular matrix for 2D stokes flow in finite differences
I have a problem by solving stokes flow in 2D by finite differences. I am using a marker and cell method, my scheme is
o-------vx1.1-------o-------vx1.2-------o
| | ...
6
votes
1answer
144 views
Schroedinger/Diffusion equation with Crank-Nicolson in Python/SciPy
I tried to make the question as detailed as possible. I have an extremely simple solver written for the Schroedinger equation but with imaginary time, which transforms it basically into the diffusion ...
13
votes
2answers
164 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
2answers
92 views
2D Schrödinger time-independent finite difference and eigenvalues
I'm learning about numerical methods to obtain the eigenvalues of a system. I have to find the eigenvalues for the time-independent Schrödinger equation but I'm having some difficulties understanding ...
1
vote
1answer
60 views
Error analysis of WENO scheme
I have three questions regarding WENO schemes
1) How to actually compute the smoothness indicators $\beta_j$ for required order of polynomial? Any reference which explains the algorithm will be ...
5
votes
3answers
122 views
What are the negatives of using higher order finite diference schemes?
I was looking at this wikipedia page:
http://en.wikipedia.org/wiki/Finite_difference_coefficient
It is a lists of higher order finite difference approximations, is there any negatives in using these ...
4
votes
2answers
103 views
Testing 1D Poisson Solver
I'm trying to test a simple 1D Poisson solver to show that a finite difference method converges with $\mathcal{O}(h^2)$ and that using a deferred correction for the input function yields a convergence ...
2
votes
3answers
106 views
Open boundary conditions with the advection-diffusion equation
Following on from my previous equation I'm would like to apply open boundary condition to the advection-diffusion equation (with reaction term),
$$ \frac{\partial \phi}{\partial t} = ...
13
votes
1answer
223 views
Strange oscillation when solving the advection equation by finite-difference with fully closed Neumann boundary conditions (reflection at boundaries)
I am trying to solving the advection equation but have a strange oscillation appearing in the solution when the wave reflects from the boundaries. If anybody has seen this artefact before I would be ...
1
vote
1answer
61 views
Effective “thickness” of finite-difference material elements
I have a 3D finite-difference formulation (for time-harmonic Maxwell using a standard Yee cell grid.) A "perfect electrical conductor" condition can be implemented by enforcing the desired electric ...
4
votes
2answers
503 views
Writing the Poisson equation finite-difference matrix with Neumann boundary conditions
I am interested in solving the Poisson equation using the finite-difference approach. I would like to better understand how to write the matrix equation with Neumann boundary conditions. Would someone ...
2
votes
1answer
113 views
FEM for non-divergence form elliptic equation
The FEM is usually used with a weak form of PDE. But for the non-divergence form elliptic operator
$$
-a_1(x,y) \frac{\partial^2}{\partial x^2} - a_2(x,y) \frac{\partial^2}{\partial y^2}
$$
or ...
4
votes
2answers
95 views
I'm having trouble debugging multigrid. What to do?
I've spent far too much time coding and debugging multigrid. While I clearly can't post all of my code as it would be silly to ask someone to go through all that code, is there anything I should pay ...
4
votes
2answers
87 views
Regularization of a discontinuous source term in an elliptic pde
Suppose I'm solving $$\frac{d}{dx}\left(K(x)\frac{du}{dx}\right)=f \text{ in }\Omega,$$ $$u=g \text{ on } \partial\Omega$$where $K(x)$ is smooth and
$$
f(x) = \left\{
\begin{array}{ll}
...
2
votes
1answer
196 views
Meaning of CFL condition on parabolic problems
I've been studying this FEM theory and for the parabolic problems, there's the analysis of stability of the $\theta$-method.
I followed the analysis and they get this CFL (Courant-Friedrich-Lewy) ...
6
votes
0answers
75 views
How to solve a compressible flow through a porous medium with a changing inlet concentration
I'm trying to solve the transient behaviour of a compressible flow through a porous media (porosity is $\epsilon$). Where the fraction of B in A ($ \phi$) changes with time. The equations that I found ...
12
votes
4answers
385 views
uniform vs. non-uniform grid
It is probably a student level question but I can't exactly make it cleat to myself. Why is it more accurate to use non-uniform grids in the numerical methods? I am thinking in the context of some ...
5
votes
1answer
178 views
Nonlinear wave equation - Finite element or finite difference
I would like to know the which is more advantageous when it comes to solving nonlinear hyperbolic equations, Finite Element or Finite difference methods? Which method will be better in capturing ...
1
vote
1answer
90 views
FEM oscillations for polynomials of degree 1
I have the following eliptic 1-D problem
$$-\mu u'' + \beta u' = 1$$
$$u(0) = u'(1) = 1$$
where $\mu = 10e^{-5}$ and $\beta = 1$. For this specific problem I am using the following space steps ...
7
votes
4answers
167 views
When we use Bernstein polynomials in application
When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple ...
4
votes
1answer
174 views
How to design good finite difference schemes? [closed]
In principle, I know finite differences. In university, we discussed it and derived consistency and boundary conditions.
But I am still left with a big question.
How to design a good finite ...
1
vote
2answers
89 views
how to approach time zero when the equation is not defined at that point
Some background: define a process $Y_t=\frac{1}{t}\int_0^tX_sds$, where $X_t$ is a standard Brownian motion. Then I define a function $u(t,X_t,Y_t)$ and require it to be a martingale. Thus, by ...
2
votes
0answers
85 views
Orthogonal vs general curvilinear coordinates
Solutions to PDEs over irregular domains can be computed using the finite difference method by the introduction of so called body fitted coordinate systems where the coordinate lines are aligned to ...
3
votes
1answer
369 views
Laplace's equation problem in Polar Coordinates (Edit)
Is there public code in Matlab for solving the Laplace equation in polar coordinates in a circular domain?
I tried a lot but my level of matlab and mathematica is not good enough, but still not ...
1
vote
1answer
88 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 ...
1
vote
0answers
64 views
Resampling of values between body fitted and cartesian grids
Assuming there exist two 2D grids on the same domain, one of them is a cartesian one while the other is curvilinear (body fitted regular grid, with curved coordinate lines).
I am looking for a way to ...
4
votes
4answers
291 views
how to test convergence of the solution vector
I am using some finite difference algorithm to solve the problem of a parabolic equation. Reading the Leveque's book on finite differences he suggests to test convergence of the method by considering ...
3
votes
2answers
107 views
Effect of boundary condition on the local error
Any error analysis is based on the Taylor expansions. So, if I take a finite difference scheme, I can calculate the value of the function at any point using the known value at another node via Taylor. ...
6
votes
0answers
201 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
157 views
Numerical solution of fractional integro-diffrential equ. using collocation method?
problem comes from
"Numerical solution of fractional integro-differential , equations by collocation method , E.A. Rawashdeh, Department of Mathematics, Yarmouk University, Irbid 21110, Jordan"
...
5
votes
1answer
62 views
Linearized implicit time stepping
Consider the general FD implicit time stepping scheme
$\frac{x_{t+1} - x_t}{\Delta t} = f(x_{t+1})$,
where $x$ is the vector variable of interest and $f$ is some function, generally non-linear.
...
8
votes
2answers
270 views
Finite difference scheme for “wave equation”, method of characteristics
Consider the following problem
$$ W_{uv} = F $$
where the forcing term can depend on $u,v$ (see Edit 1 below for the formulation), and $W$ and its first derivatives. This is a 1+1 dimensional wave ...
4
votes
2answers
409 views
Implementing a finite difference method in Mathematica
I am trying to iterate the following equation
$$
x_{k}(n+1)=x_k (n)-\epsilon (x_{k+1}(n)-2x_k(n) +x_{k-1}(n))+\sqrt{\epsilon}\; \eta_{k}(n)
$$
where $n$ denotes which time step I'm on and $k$ is the ...
7
votes
3answers
415 views
Boundary conditions for the advection equation discretized by a finite difference method
I am trying to find some resources to help explain how to choose boundary conditions when using finite difference methods to solve PDEs.
The books and notes which I currently have access to all say ...
5
votes
2answers
181 views
Interpolation schemes to move data between cells and nodes
I work on non-graded quadtree grids where the entire grid is a hierarchy of cells specified using a quadtree data structure, where, in general, there is no constraint regarding the relative size of ...
7
votes
1answer
371 views
Finite difference coordinate transformation for spherical polar coordinates
I have part of a problem that is described by the momentum conservation equation:
$\frac{\partial \rho}{\partial t} + \frac{1}{\sin\theta} \frac{\partial}{\partial \theta}(\rho u \sin \theta) =0$
...
1
vote
2answers
96 views
Smoothing the diffusion coefficient to improve convergence
I have been reading a book by Thomee and he considers the case of $u_t=(au_x)_x$, for the case of $a$ possibly being discontinuous. Then he says that the problems with convergence might occur, and ...
