Questions tagged [finite-difference]

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

Filter by
Sorted by
Tagged with
3
votes
2answers
883 views

Finding shortest path in a time/distance map

I get a distance map output after using a Fast Marching Method. The PDE involved is the Eikonal equation which take the form : $$\begin{cases} c(x).|\nabla u| =1\\ u(x) =\phi(x) ...
3
votes
1answer
615 views

Problem in Discretizing Convection-Diffusion-Reaction equation

I'm trying to solve the Convection-Diffusion-Reaction (CDR) equation on a rectangular domain, using cylindrical coordinates and Finite Difference Methods (FDM) (this approximates a flow reactor). ...
3
votes
1answer
162 views

Definition of TV in TVD finite difference methods

TVD (total variation diminishing) finite difference methods that produce non-oscillatory solutions are based on the total variation. In LeVeque's book the total variation of a function $q(x)$ is ...
3
votes
2answers
447 views

Simple CFD method

I am looking for a simple method to compute potential flow (then non-viscous) around an obstacle. The method I am looking for must NOT use coordinate transformation, panel methods, finite element ...
3
votes
1answer
299 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 ...
3
votes
2answers
324 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" $D^...
3
votes
1answer
72 views

Maintain unitary time evolution for a nonlinear ODE

I want to solve a nonlinear ODE of matrix $A(t)$ $$\mathrm{i}\dot A = A(t)M(t),\:\mathrm{with}\: M(t)=A^\dagger(t)H(t)A(t)$$ where $H(t)$ and hence $M(t)$ are Hermitian. Therefore, I presume the time ...
3
votes
1answer
806 views

Finite difference method basic implementation on Octave

Trying to study the error of FDM for a second order derivative versus step size I calculated the coefficients and validated them, but the output has errors for small step sizes. The function in ...
3
votes
1answer
154 views

interpolation 2D irregular nodes

Given a 2D irregular spaced data like shown in the figure, I would like to know how to find derivatives at '*' by interpolating the values at 'o'. Does lagrange 2D interpolation work at irregular ...
3
votes
1answer
208 views

Is this finite difference approach correct?

I am solving incompressible 2D Navier-Stokes equations with zero y-component velocity. The geometry is a simple 2D pipe of a length $L$ and diameter $W$ and there is only two boundary conditions: Non-...
3
votes
1answer
372 views

Are the Finite Difference and Finite Volume Methods different after the application of the Gauss Divergence theorem on the FVM?

I have a rudimentary question about the differences between finite difference (FDM) and finite volume methods (FVM). In FDM we concentrate on the nodes (points) in space while in FVM we concentrate ...
3
votes
1answer
5k views

Why a finite difference scheme would give second order of accuracy in norm L2 but 1.5 with L1 (while 1 with Linf)?

My finite difference scheme for the 2D Euler equations is second order accurate in theory, since all the terms are second order accurate, with the advective terms being third order. So I expect a rate ...
3
votes
2answers
829 views

Get symmetric Finite Difference matrix in non Laplacian settings

I would like to solve a system of differential equations $u+\nabla(\nabla\cdot u)=f$ or in more detail $a+\partial_t^2a+\partial_t\partial_xb+\partial_t\partial_yc=f$ $b+\partial_x^2b+\partial_x\...
3
votes
2answers
216 views

Finite element: handle discontinuity/abrupt interface

I am using Galerkin's method to set up some pde solver for my problems(1D/2D) at hand. Some abrupt material interfaces do exist, so far, what I did regarding this is simply: 1) no element across the ...
3
votes
1answer
2k views

How to solve the advection equation in 2 dimension using the Crank-Nicolson method?

I've an equation like this to solve with the crank-nicolson method $$U_t -\frac{y}{2} U_x + \frac{x}{2}U_y = 0,$$ where $x$ and $y$ are: [-2,5:2,5] and the time $T$ ...
3
votes
1answer
129 views

Comparison between FEM and FDM methods for flow simulations

What are the main differences between finite element and finite difference approach for incompressible flow simulations? I have a vague idea about how FE methods rely on minimizing the residual over ...
3
votes
1answer
211 views

Numerical Lax-Wendroff scheme order of convergence on Burgers equation

I was suggested to move that question here. The question to be as follows. Statement of the problem Is it possible to achieve the second order of convergence (OOC) of Lax-Wendroff (LxW) scheme ...
3
votes
2answers
146 views

In numerical methods, eg, finite differencing approaches, does there exist convergent schemes that are not both consistent and stable?

In a book that our course is following this semester, the theorem given is only in one direction: if the scheme is both consistent and stable, then the scheme is convergent. However, since this ...
3
votes
1answer
950 views

1-D incompressible unsteady Couette Flow Explicit finite differece CFD

I am currently following J.Anderson Jr.'s CFD with basic application and I came into some troubles while coding for my very first CFD problem. As the title suggests I am solving an incompressible ...
3
votes
1answer
143 views

Boltzmann equation and equilibrium distribution function

The free-streaming term in stationary 1D Boltzmann equation satisfies the equilibrium distribution function, that is: \begin{equation} \text{Fr}\{f\} := v(k)\dfrac{\partial f(x,k)}{\partial x} - \...
3
votes
2answers
128 views

How does constraint resolution affect the stability/accuracy of numerical integration?

I understand some basic analysis techniques (local truncation error, global error, zero-stable, absolute stable, etc.) of numerical integration. But I find it hard to apply these techniques in ...
3
votes
1answer
4k views

Finite differences scheme for 2D advection equation

I'm trying to study with a finite difference method the 2D advection equation with a space-dependant flow. Taking a function $f(x,y,t)$ solution of the equation : $$ \partial_t\,f+\nabla(\textbf{v}\,...
3
votes
1answer
398 views

Finite difference equations versus boundary integral equations for elliptic pdes

In certain cases, boundary integral methods are preferred for elliptic partial differential equations as opposed to finite difference methods. For instance, for solving the Poisson equation in a ...
3
votes
1answer
135 views

Dirichlet Boundary Condition finite difference method using sparse-matrix $Ax = b$ system

I am trying to solve the boundary value problem for heat equation: $$ u_{xx} + u_{yy} = f(x,y) $$ where the solution $u(x,y) \in [0,1] \times [0,1]$ and the Dirichlet boundary condition $u(x,y) = ...
3
votes
2answers
159 views

Damped Harmonic Oscillation. Efficient algorithm to find the parameters resulting in threshold oscillation amplitude

Let's assume, that we have damped harmonic oscillation of a body in the form of a cone, immersed in a liquid. Equilibrium condition of the body is: $$m\overrightarrow{a} = \overrightarrow{F_\text{...
3
votes
2answers
1k views

Implementing no-flux boundary condition reaction-diffusion PDE

I'm having trouble figuring out how to implement boundary conditions for this problem: \begin{align} \frac{\partial n}{\partial t} &= D_n\nabla^2n - \nabla\cdot\left(\frac{\chi}{1+\alpha c}n\nabla ...
3
votes
1answer
251 views

Stability Criterion for this Explicit Scheme

I am solving an unsteady flow using the dual-time Navier-Stokes equation in which I write my momentum equation as: $$\frac{\partial u}{\partial \tau} + \frac{\partial u}{\partial t} + \frac{\partial u^...
3
votes
1answer
1k views

How to calculate dispersion relation from a Finite Difference (FD) wave simulation

I have a python code that calculates the solution of the inhomogeneous acoustic wave equation for a 2D medium with any velocity and source configuration. It was implemented using Finite Differences ...
3
votes
1answer
408 views

Finite difference scheme for 2D sound propagation

I am simulating the sound wave propagation in non-rectangular and asymetric spaces using finite-difference method. I presume linear acoustics equations to be enough (i.e. $\Box p = 0$, $\Box \vec{v} = ...
3
votes
1answer
2k views

Solving a Nonlinear BVP using Finite Difference Method

I am trying to write a code to solve a nonlinear BVP using the Finite Difference Method. The BVP is: $(T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0$ ...
3
votes
1answer
166 views

Books and references on implementing finite difference codes for PDEs

Are there any good books or references on implementing finite difference methods for PDEs? Specifically, I'm looking for something comparable to Gockenbach's book Understanding and Implementing the ...
3
votes
0answers
55 views

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 ...
3
votes
0answers
51 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
votes
0answers
82 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
votes
0answers
89 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
56 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
86 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/...
3
votes
0answers
114 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
38 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
161 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
73 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
108 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
497 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
94 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
241 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
445 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
325 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
271 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
462 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
210 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 ...

1
3 4
5
6 7
15