Questions tagged [finite-difference]
Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.
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Coupled Partial Differential Equations
I'm trying to solve the following system of coupled differential equations, the two-temperature model for $e$ = electrons and $l$ = lattice.
$$
\rho_{e}C_{p,e}\frac{\partial T_{e}}{\partial t} = k_{e}\...
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How can I get more accurate electric scalar potential in 2D closed box?
I am trying to use poisson equation to plot the electric scalar potential in close 2D space. The details in in this video
and this one
The following in written in Matlab for quick prototype.
...
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Asking advice for implementation of Conservative Finite Difference Scheme for numerically solving Gross-Pitaevskii equation
I am trying to numerically solve the Gross-Pitaevskii equation for an impurity coupled with a one-dimensional weakly-interacting bosonic bath, given by (in dimensionless units):
\begin{align}
i \frac{\...
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Imposing higher order finite difference schemes for boundary value problems on a finite interval
I have some questions. I'm going to assume everything is in 1d with a Laplacian operator. If I discretize the Laplacian operator using $p = 2a+1$ grid points with periodic boundary conditions, I ...
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Deriving order of accuracy and interpreting a given discretization scheme when underlying method ( finite difference/volume) not known
If a spatial grid is given with time levels like this:
to solve the following model problem
Now consider the following discretization schemes:
Scheme 1
Scheme 2
Usually, to determine order of ...
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Implicit-Explicit Operator Splitting Scheme
I am trying to solve the 2D advection-diffusion equation in cylindrical coordinates:
$$
\frac{\partial c}{\partial t} = D\left(\frac{\partial^2 c}{\partial r^2} + \frac{1}{r}\frac{\partial c}{\partial ...
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Numerically solving the Advection-diffusion equation with no-flux boundary condition leads to violation of mass conservation
I am trying to solve numerically the advection-diffusion equation of the following form
$$\frac{\partial C}{\partial t}=\alpha\frac{\partial^2 C}{\partial x^2}+\beta \frac{\partial C}{\partial x}$$
...
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"Cleanest, most professional" approach to implement a finite differences scheme in dimension greater than two
Coding a finite difference algorithm in 1D does not require a complex mesh. In higher dimensions, you would need a mesh and its connectivity graph to compute the differential operators. Finite ...
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Vector poisson equation in cylindrical coordinates. What's wrong?
I am trying to solve this equation:
$$
\frac{\partial^2 A}{\partial r^2} + \frac{1}{r}\frac{\partial A}{\partial r} - \frac{A}{r^2} + \frac{\partial^2 A}{\partial z^2} = -g
$$
This is basically the ...
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Can domain decomposition methods also be applied to linear systems resulting from finite difference discretizations?
Question
In general, do domain decomposition methods (DDMs) require a linear system of equations $Au = f$ to be formed by finite element discretization/method (FEM) of a PDE? Or could one simply use a ...
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Are there necessary conditions for SOR algorithm to converge when used for 2D discrete Poisson problem, with PBC, besides solvability condition?
I am trying to solve a 2D poisson problem that is supposed to represent diffusion of chemicals on a grid: $\nabla^2 R_{ij}=f_{ij}$. I am discretizing the problem with the standard central ...
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numerical schemes for 1D PDE: for smaller grid size there is an increased roundoff error, larger size more truncation, so sweet spot in between?
I had a discussion with a colleague today. He claimed that usually for a general numerical scheme for solving a general 1D PDE, for smaller grid size there is an increased roundoff error because of ...
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How to quantify the numerical diffusion term in a second-order upwind advection scheme?
In the first-order upwind scheme, numerical diffusion can be quantified as:
$$\frac{dT}{dt} = -w\left(\frac{dT}{dz}\right) + \left(\frac{wdz}{2}\right)\left(\frac{d^2T}{dz^2}\right)$$
For Lax-Wendroff,...
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Why does the FDM give a correct solution to a PDE with a discontinuous initial condition?
I was solving the dimensionless wave equation:
$$ u_{xx} = u_{tt} \tag 1$$
with the initial conditions:
$$ u(x,0) = 0 \tag 2 $$
$$ u_t(x=0,0) = v_0 \tag 2 $$
$$ u_t(x>0,0) = 0 \tag 3 $$
and the ...
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How to calculate the force of solid applied by fluid? Using finite difference method, DNS, staggered grid, SIMPLE algorithm, immersive boundary
Problem
I am using finite difference method to solve classic problem of flow around cylinder, for validation of my group's immersive boundary method.
The common way to validate numerical method is ...
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