Questions tagged [finite-difference]
Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.
160
questions with no upvoted or accepted answers
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votes
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85 views
Evolutionary dynamics in vascularised tumors, PDE-ODE coupled system
I have to solve the following PDE-ODE system
$$ \displaystyle{\partial_{t} n = \bigl[a(s) - b(s)(y - h(s))^{2} - d\int_{\mathbb{R}} n \, dy \; + \; \beta \, \partial^2_{yy} n \;}
\\\\
\...
5
votes
0answers
82 views
Numerical methods for the continuity equation with Sobolev vector field
Consider the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$.
...
5
votes
0answers
160 views
Is it generally unstable to use in multidimensional simulations finite difference schemes with higher orders than 2?
I'm part of a team trying to generalize a 1D Advection-Diffussion-Reaction code we inherited by extending it to 2D by using dimensional splitting, i.e. solving advection and diffusion for x and y ...
5
votes
0answers
729 views
Can Gauss-Seidel/SOR (preconditioned?) be applied to a non-diagonally dominant matrix?
After applying finite difference method to a Laplace/Poisson problem always arises a diagonal dominant system of equations that can be solved with Gauss-Seidel or SOR methods. If the original PDE does ...
5
votes
0answers
115 views
How to choose a stable PML for pseudo-spectral method with strongly varying velocity
My friend was working on this, and he asked me about the stability of PML while applying on pseudo-spectral method, I believe his concern was how to choose the difference(if the difference should be ...
4
votes
0answers
189 views
A good 2D finite difference for the continuity equation
How could I go about solving the continuity equation below in 2D?
$$\frac{\partial \rho}{\partial t} + \nabla \cdot \left(\rho u\right)=0$$
I saw that a similar question was posted here: A good ...
4
votes
0answers
163 views
How can I solve the wave equation for a circular rod in cylindrical coordinates using finite differences?
I have a problem with the stability of finite difference method for the wave equation in cylindrical coordinates.
the equation is:
$$
\frac{\partial^2 \omega_n}{\partial r^2}+\frac{1}{r}\frac{\...
4
votes
0answers
784 views
How to implement boundary conditions on Finite Difference WENO5 scheme for the Euler equations
I'm implementing a Finite Difference WENO5 with Lax-Friedrich flux splitting on a uniform, structured grid to solve the 2D Euler equations of fluid dynamics on a rectangular domain in cartesian ...
4
votes
0answers
224 views
Stability analysis for a hyperbolic PDE on staggered grid
I am trying to understand the stability of a finite difference equation on the staggered grid.
I could understand the Von Neumann stability analysis for the collocated grid for a simple acoustic ...
4
votes
0answers
116 views
The numerical solution of a (very ugly) set of integro-diferential equations
I've been having fun playing around with a simple model for reaction in a co-flow fluidized bed (I'm afraid I'm a chemical engineer, not a computer scientist). Unfortunately, simple physical ...
4
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0answers
1k views
CFL Condition and Convection Diffusion Equation in 2D
I am solving the convection-diffusion equation in 2D using Finite Differences with the $\theta$ scheme. The velocity of the fluid and the diffusion coefficient is low in my case (in the range of $10^{-...
4
votes
0answers
458 views
Numerically calculating the divergence of a set of oriented points
Say I have a set of oriented points at locations $\vec{v_i}$ with each some direction $\vec{n_i}$, in practice they represent the normals of some surface that has been non uniformly sampled. How would ...
4
votes
0answers
915 views
2D wave equation with Mur boundary condition - setting up the RHS and solving (time-steps)
I am trying to solve a 2D wave equation implicitly using FD with central approximations with the following boundary conditions
$$\begin{align}
&u=2\sin\left(\frac{2\pi}{5}t\right)\quad \text{at }...
4
votes
0answers
147 views
numerical analysis of a partial integro-differential equation
I have to numerically solve a nonlinear partial integro-differential equation. This is my equation,
$$\frac{\partial y(x,t)}{\partial t}=\int_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} \frac{\...
4
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0answers
1k views
Python - calculation time derivative and laplacien by finite differences
I would like to determine a temporal derivative and a Laplacian by the finite differences method to solve the heat equation in a 1-dimensional case. My aim is to get the sources term that is why I ...
3
votes
0answers
41 views
Spectral solver on em-pic
I'm recently studying for the spectral solver to implement EM-PIC code. I read an article and have some questions.
Many PIC codes uses spectral solver to overcome numerical artifacts on FDTD.
In the ...
3
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0answers
57 views
What is the reason for this finite-difference high errors on non-uniform grid?
tl;dr
Using a Taylor-matched method to find coefficients for the discretized equation $ \mathbf{A} \vec{f}'' = \mathbf{B} \vec{f} $, a Fortran code has been implemented to find the second derivative ...
3
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0answers
85 views
numerical instabilities in Fluid Dynamics, Finite Element Method
I'm looking for references to understand where the numerical instabilities come from in hydrodynamics in general, and notably when the Péclet number: $Pe>1$. I'm using the finite element method.
...
3
votes
0answers
53 views
Use of non-typical values of $\theta$ in theta-methods
The theta-method is a popular solution for solving time-transient PDEs (or ODEs), which consists of solving the general equation for each time step:
$$
\frac{u^{n+1} - u^{n}}{\Delta t} + (\theta f(u^{...
3
votes
0answers
103 views
WENO5 scheme in a staggered grid
I'm trying to use the finite-difference WENO scheme to solve the 2D density conservation law with axial symmetry (coordinates $r,z$):
$\frac{\partial \rho}{\partial t}+\nabla \cdot (\rho \vec{v}) = \...
3
votes
0answers
57 views
How to implement register blocking for 3D finite-difference stencil computations
I am in need to optimize the performance of the 3D solver that integrates a system of equations for seismic wave propagation.
Unsurprisingly, the function that implements the finite-difference stencil ...
3
votes
0answers
36 views
Guide for finite-difference schemes for Hamilton-Jacobi-Bellman Equations
I need to solve a simple, low-dimensional Hamilton-Jacobi-Bellman equation.
Is there a simple guide for doing this numerically using finite-difference schemes? I found a few research articles ...
3
votes
0answers
124 views
Factorize laplacian in terms of first derivative matrix
I am trying to factorize the following Laplacian matrix in terms of $ D^TD$, D is the first derivative matrix.
The tridiagonal form of the secon derivative matrix using Neumann boundary condition is ...
3
votes
0answers
67 views
Solving numerically a linearized system of elliptic (?) Navier-Stokes equation (Shallow Water Derived)
For my PhD Thesis, my advisor asked me to build a solver inspired from the article "Optimal Control Theory Applied to an Objective Analysis of a Tidal Current Mapping by HF Radar, J-L Devenon, 1989". ...
3
votes
0answers
101 views
How to account for the interface between two different phases in a discretized diffusion model?
I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a ...
3
votes
0answers
408 views
Neumann-Neumann boundary intersection
I am trying to solve a Vertex Centered, Finite Difference, Poisson equation $-\nabla^{2}u=f$ with Dirichlet boundaries on the left and bottom and Neumann boundaries on the top and right using ...
3
votes
0answers
89 views
How to optimize for decay constant in exponential-like function?
I've got a data set of points $M_O .. M_N$ for time points $t_0 .. t_N$, where $N$ is approximately 10-20, and the spacing of time is not uniform (i.e., $t_{i+1}-t_i$ is not constant for all i). It is ...
3
votes
0answers
215 views
Understanding Davis artificial viscosity
I'm solving 2D Euler's equations in Cartesian coordinates:
$$
\frac{\partial U}{\partial t} + \frac{\partial F}{\partial x} + \frac{\partial G}{\partial y} = 0, \qquad
U = \left( \begin{array}{c} \...
3
votes
0answers
357 views
Gradients of non-uniformly sampled data in 3D space
I have measurements of magnetic field on a 3d grid. My measurements are distributed on four x-y planes similar to what is shown in the image below. The measurements roughly follow a Cartesian grid but ...
3
votes
0answers
280 views
Corner Transport Upwind for Linear Advection in Arbitrary Velocity Field
I need to implement a 3D version of the Corner Transport Upwind (CTU) finite volume method (in python); and so I've been reading Leveque, "Finite Volume Methods for Hyperbolic Problems" which I think ...
3
votes
0answers
205 views
matplotlib contourplot for $\log z$ in the Complex Plane $\mathbb{C}$
I tried using Python's matplotlib on the logarithm and here is what I got, a kind of starburst pattern. Since the angle jumps between $\theta = 0$ and $\theta = 2\pi$, contour assumes there is a ...
3
votes
0answers
418 views
Boundary equations for constant right hand side in Poisson equation (FD)
I am setting up to solve a 2D Poisson equation $\nabla^2u = T$ by the finite difference method.
I am using the multigrid method. On the coarsest mesh, where the grid size is 2x2, I want to set up ...
3
votes
0answers
188 views
Computing accuracy of my finite difference scheme for uniform grid on a non-uniform grid
I have a ADI finite difference scheme for the 2D Navier-Stokes equations that uses a second order accurate (central) approximation for the advective terms. I am ignoring the diffusive terms for now. I ...
3
votes
0answers
305 views
Efficient assembly of finite element matrix(coupled equations case)
I noticed this post, where spalloc and sparse are recommended for efficient assembly in Matlab. I personally use sparse ...
3
votes
0answers
343 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 ...
2
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0answers
35 views
How does the “Stable Fluids” algorithm by Jos Stam relate to the SIMPLE and PISO algorithms?
The "Stable Fluids" paper (*) by Jos Stam starts by acknowledging that "Our method cannot be found in the computational fluids literature, since it is custom made for computer graphics applications. ...
2
votes
0answers
40 views
How to determine the finite difference coefficient matrix in 2D with periodic BC?
I'm solving a PDE in matlab using ode15s, and since the spatial dimension is 2, and number of variables grow large very quickly, I need to supply the structure of ...
2
votes
0answers
29 views
Convergent Finite Difference Scheme for Parabolic Equation
Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$
where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$.
In Strikwerda's book, the ADI scehme \begin{align*}
\left( 1 - \frac{...
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votes
0answers
51 views
Numerical errors due to terms of the form $\frac{1}{r}$ (r goes to 0 at the boundary) while using finite difference method
I am trying to solve a system of differential equations using finite difference method.
There are few terms of the form $\frac{A(r)}{r}$, both $A(r)$ and r go to zero at the boundary. Analytically ...
2
votes
0answers
126 views
Implementation of boundary conditions for 1D Euler equations
I'm trying to solve 1D Euler equations with gravity in spherical coordinates using a finite-difference TVD MacCormack method on a non-uniform grid of $N$ components, following the method provided in ...
2
votes
0answers
43 views
Verification on pressure predictor method for CFD code
I have developed a python code for a lid-drive cavity model. However, my results are not converging. The algorithm of my code looks like this:
Euler Momentum Equation looks like this:
$$\frac{u^{n+1}...
2
votes
1answer
154 views
Solving nonlinear PDE with finite difference based on Newton-Krylov
I am now working on solving MHD equations with finite difference method, which include nonlinear equations:
$$
\frac{\partial\rho}{\partial t}+\nabla\cdot\left[\left(\rho_0+\rho\right){v}\right]-\...
2
votes
0answers
44 views
How to determine the order of convergence of the Euler-Maruyama method?
This question is originally posted in Quant.StackExchange but has been unanswered for some time so I ask in here.
To make this simple let us consider the Geometric Brownian Motions (GBM).
My ...
2
votes
0answers
73 views
Numerical solution to N-dimensional diffusion on simplex?
Assume I have a system of at least (but generally only) $N+1$ points in an $N$-dimensional space ($N > 3$ is possible). At each of these points $x_i, i=1,...,N+1$ I know an initial potential/...
2
votes
0answers
60 views
Unusual boundary conditions on Matlab
I'm trying to solve the following PDE by Matlab,
$$
u_t-\Delta u = 0, \quad \text{in}\quad \Omega\times (0,T)
\tag{1}
$$
$$
u_t-\Delta_\Gamma u + \partial_\nu u=0,\quad \text{on}\quad \Gamma\times(0,...
2
votes
0answers
87 views
Can I take advantage of a nearly banded A in AX=b?
I am working on a 1D drift-driffusion problem in a finite-difference (FD) approach.
I hade 3 equations per node ($3N$ in total): electron continuity $E_i$, Poisson $P_i$, hole continuity $H_i$. With ...
2
votes
0answers
95 views
Neumann boundary conditions in the Maccormack scheme
I am trying to solve the viscous Burger equation
$$
\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \mu \frac{\partial^2 u}{\partial x^2}
$$
with Neumann boundary conditions. I am
...
2
votes
0answers
47 views
Finite difference Neumann boundary conditions: uneven weighting of edge nodes?
Originally asked this on math.stackexchange, but I figure it's also appropriate here. I'm reading through some finite difference code for a diffusion equation and came across something odd for the ...
2
votes
0answers
66 views
Finite differenced eigenvalue prob. of inhomogeneous boundary conditions?
I am basically asking about eigenvalue problems of differential equations using some finite difference method (FDM). Usually the system is subject to some boundary conditions (BC), e.g., Dirichlet or ...
2
votes
1answer
161 views
Is there a simple way to avoid carbuncles for FD WENO methods?
I have implemented finite-difference WENO scheme for Euler equations (with some variants - WENO-JS, WENO-Z, WENO-M, different flux splitting). It works well, but have problem with so-called carbuncles ...
