Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [finite-difference]

Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.

5
votes
0answers
110 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 ...
5
votes
0answers
110 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
134 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
97 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 ...
4
votes
0answers
122 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
178 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 ...
4
votes
0answers
541 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
169 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 ...
4
votes
0answers
183 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
299 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
143 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 ...
4
votes
0answers
141 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
votes
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
31 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
56 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
84 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
112 views

How one could choose the value of viscous coefficient for obtaining stable solution of Burgers' equation?

Burgers' equation is a fundamental PDE used in various fields such as number theory, gas dynamics, heat conduction, elasticity, etc. It is crucial especially for developing numerical models for ...
3
votes
0answers
216 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
907 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^{-...
3
votes
0answers
144 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
690 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 }...
3
votes
0answers
358 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
154 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
262 views

Efficient assembly of finite element matrix(coupled equations case)

I noticed this post, where spalloc and sparse are recommanded for efficient assembly in Matlab. I personally use sparse assembling for simple cases. However, when it comes to the case of coupled PDE, ...
2
votes
0answers
48 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 ...
2
votes
0answers
60 views

2d wave equation with finite differences blowing up

I am (naively) trying to solve the 2d wave equation with finite differences. But the system blows up instantly. For simplicity I set the constant $c=1$, then I am left with $$\Delta u =u_{tt}.$$ I ...
2
votes
0answers
82 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 ...
2
votes
0answers
27 views

Numerical stability while modeling wave equation in staggered grid

I have modeled a simple wave equation given by: \begin{align} & \begin{cases} u_t = v_x \\ v_t = u_x \end{cases} \end{align} Boundary conditions given on interval $M=[-1,1]$ by: \begin{align} u(...
2
votes
0answers
27 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
36 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
56 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
0answers
146 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} \...
2
votes
0answers
110 views

Finite difference aproximation - Darcy law

I am solving following problem: Filtration of water can be described in bi-dimensional case by $$- \partial_x(K(x,y) \partial_x u ) - \partial_y (K(x,y) \partial_y u ) = 0, $$ where $u$ - water ...
2
votes
0answers
98 views

Calculating theoretical order of accuracy of least squares fit advection scheme

I'm familiar with finding the order of accuracy using von Neumann analysis for finite difference schemes formulated using Taylor series expansions. But is there a similar technique for finding the ...
2
votes
0answers
341 views

Finite difference scheme for solving nonlinear least-squares problem

I am dealing with following problem: $$ \min_{u,\gamma}\Bigg\{ \frac{1}{1000} \iint_{S_2} {\gamma (x,y)^2 dxdy} + \iint_{S_2} {[u(x,y) - u_0 (x,y)]^2 dxdy} + \iint_{S_2} {[\Delta u(x,y) - \gamma (...
2
votes
0answers
115 views

Rank deficient Jacobian in discretized periodic solutions to autonomous ODE

I'm trying to numerically find periodic solutions to different systems of autonomous nonlinear ordinary differential equations. I decided to use a finite difference scheme and solve the resulting ...
2
votes
0answers
211 views

1 D Diffusion equation FDM with different layers

I'm trying to solve this particular equation $\frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \big[D_{i}(x)\frac{\partial u}{\partial x} \big] + S(x,t)$ where the $i$ index denotes ...
2
votes
0answers
382 views

What is the difference between SSPRK3 and RK3 time discretization methods?

Here, SSPRK3 refers to third order strong stability preserving Runge-Kutta and RK3 refres to regular third order Runge-Kutta method. The meaning of the method is obvious from the name. However there ...
2
votes
0answers
60 views

2nd order differences for modelling a physically discrete phenomenon?

Consider the diffusion of some unknown $R$ on a physically discrete, 1-D ring: x---------x---------x---------x n = 0 1 2 3 We let ...
2
votes
0answers
376 views

2D Neumann Conditions on Irregular Domain

I would like to model the 2D diffusion equation with Neumann BC's inside the following egg-shaped domain: IMG http://i62.tinypic.com/33m982r.jpg I would like to use the finite difference method with ...
2
votes
0answers
383 views

Solving the Helmholtz equation numerically

I am trying to solve the Helmholtz equation in some complex unbounded 2D domain: $$\left( \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + E \right) u(x, y) = 0$$ ...
2
votes
0answers
140 views

Why does my Finite Difference approximation not work?

I am trying to find out the magnitude of the acceleration of my object based on non-uniformly sampled 3D position data. I'm using the standard approximation of the 2nd order derivative on a non-...
2
votes
0answers
814 views

How to solve singular non symmetric poisson equation with Neumann boundary condtions?

I am trying to solve 2D Poisson equations with Neumann boundary conditions. When the mesh is uniform, Poisson equation is singular and symmetric, so the method listed in Null Space Projection for ...
2
votes
0answers
367 views

finite volume for diffusion equation with anisotropic (tensor) coefficient

Consider the scalar PDE for $u$ with Dirichlet boundary conditions: $\mathrm{div}(\mathcal{K}\nabla u) = f\; \forall x\; \in \Omega \subset R^2$, $u = 0 \; \forall \; x\;\in \partial\Omega$ ...
2
votes
0answers
71 views

Finite Difference for Hamilton Jacobi Belman

I have hjb equation where $V=V(x,t)$ and $u=u(x,t)$ $V_t + \sup(u) [A(x,u)V_x + B(x,u)V_{xx}]=0$ for $x$ in $[0,1]$ and $t$ in $[0,1]$ I have been able to successfuly resolve it numerically having ...
2
votes
0answers
283 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
votes
0answers
101 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 ...
1
vote
0answers
81 views

Finite differences for the one-phase Stefan problem

I am trying to code the one-phase, one-dimensional Stefan problem using finite differences in Matlab, similarly to what has already been done in Mathematica (see https://mathematica.stackexchange.com/...
1
vote
0answers
70 views

why I cannot find explicit finite difference for elliptic equation

Let us think on the Poisson equation $\nabla^2 u(\bf{x})=\rho(x)$ with Neumann boundary conditions, with $\bf{x}=\it (x,y)$ in 2D. Here is a stencil with central differences in both $x$ and $y$ (...
1
vote
0answers
72 views

Finite difference methods for coupled 2nd order nonlinear pdes

I have a system of coupled nonlinear PDEs that I cannot figure out how to solve in a smart way using FDM, so I was hoping someone here might have a clue. The equations go as: \begin{align*} \frac{1}{...