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Questions tagged [finite-difference-method]

Tag for the usage of "FiniteDifference" Method embedded in NDSolve and implementation of finite difference method (fdm) in mathematica.

3
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1answer
37 views

Generate coefficient array from general formula of linear equation system

This is a problem coming out in the implementation of finite difference method (FDM). Here is a toy example. Suppose we want to solve the boundary value problem (BVP) $$y''(x)=\sin(x),\ y(0)=0,\ y(...
5
votes
2answers
152 views

Solving linear coupled PDEs by FDM

I am trying to solve some linear, coupled PDEs for perturbative analysis (first order in time, 3rd order in space), for which I then plan to take the global spatial maxima of their magnitudes and plot ...
3
votes
1answer
205 views

Solution of Burgers equation with some initial data

Consider the Burgers equation $$\partial_t u + \partial_x\left(u^2/2 \right) = 0, \quad u(0, x) = u_0(x).$$ eq = D[u[t, x], t] + D[u[t,x]^2/2, x] == 0 How ...
8
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3answers
424 views

New things and limitations in Version 12 numerical differential equation solver?

This question is intended to be a place to summarize users' exemplary experience in solving differential equation with the NDSolve family in MMA’s latest version 12....
5
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2answers
374 views

Numerical methods to solve a continuity equation

What numerical methods can be used to study the initial value problem for the continuity equation where $ u = u(t, x) $ $$ u_t + \nabla\cdot(\boldsymbol b u) = 0, \qquad t \in [0,T], \quad x=(x_1,x_2)...
2
votes
2answers
238 views

NDSolve error in 2-D heat equation

I'm trying to solve the following PDE by Mathematica in 2-D case in the unit disk using polar cordinates, where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $\Gamma =\partial \Omega$ is the ...
10
votes
1answer
197 views

FiniteElement v.s. TensorProductGrid: which is reliable for Schrödinger equation with periodic b.c.?

This is a problem comes up in the discussion under this post and I think it's worth starting a new question for it. I suspect the underlying issue is the same as in this post, but not sure. Consider ...
1
vote
1answer
86 views

Solution of differential equation and then draw a graph

I have two differential equations: $da/dt = a (.3 a^{-3} + .7)^{1/2}$ and $d \tau /dt = 1/a$. The initial conditions are $t = 0$; $a = 1$ and $\tau = 0$, respectively. How can I solve the ...
1
vote
1answer
130 views

How can solve this partial differential equation (PDE) and plot?

How can plot and solve this partial differential equation in mathematica? $$ K \frac{\partial^2 T}{\partial x^2}- h (T-T_m) = \frac{\partial T}{\partial t} $$ $ Tm = 25 $ $ k= 47 $ $ h= 1.5 $ ...
7
votes
1answer
277 views

Solve PDEs with finite difference scheme by modifying NDSolve-based solver

Motivation As discussed here, NDSolve uses different difference orders for various spatial derivatives and the implicit design could cause trouble in certain cases....
2
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1answer
149 views

Using NIntegrate in Finite Difference Derivative method

I'm trying to solve a second order differential equation using the code given by @xzczd here which is based on this. What this ...
9
votes
1answer
191 views

How to find the differentiation matrix based on FEM?

Assume that I have a non-equidistant grid of $n$ nodes, as follows: ClearAll["Global`*"]; n = 10; SeedRandom[123]; nx = Sort@RandomReal[{-1, 6}, n] If I want to ...
9
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2answers
517 views

Stiff BVP of nonlinear ODE, alternative/ enhancement to shooting method

Question: I have been trying to solve this coupled ODE set. \begin{align} ( \frac{ \mu^2}{B} +1 ) \Phi^2 + \frac{1}{A} {\Phi^{\prime 2}} + \frac{1}{2}\lambda \Phi^4 - \frac{A'}{r A^...
5
votes
1answer
133 views

How to control DifferenceOrder in NDEigenvalue for an ODE?

I am trying to solve the eigenvalue problem of a 1st-order ODE system using NDEigenvalue. It should be finite difference method for ODE. And I want to tune the the ...
2
votes
1answer
101 views

Oscillations on solution of finite difference equation

In a previous post on the solution of an ODE with a boundary conditon at infinty I had some excelent help from xzczd and am now returning with a further problem along the same lines. I have used the ...
1
vote
1answer
101 views

How to invert Differences[list, order] with order >1?

Differences order 1 can be invert using FoldList[], but does not work with higher orders. For Example ...
3
votes
1answer
394 views

Schemes for nonlinear advection equation

I am working in the traffic flow problem using the Lighthill-Whitham-Richards model together with the Greenshields equation. The equation of that model is this: $$ \frac{\partial\rho}{\partial t}+v_{...
3
votes
2answers
292 views

Why ODE's naive finite difference matrix works well for different boundary conditions

We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. The naive way to write down the matrix of the differential operator is like the following, ...
0
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0answers
281 views

Derivative of list of points: DifferentiatorFilter versus DerivativeFilter

I need to calculate the derivative from a list of experimental points. Mathematica has two functions which seem to do this: DifferentiatorFilter and DerivativeFilter. Which one is better for ...
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vote
0answers
64 views

Optimizing finding the offset diagonals for a 2D Finite Difference Method

It is possible to collapse a conventional 2D PDE (in our case the Schrödinger equation) into one dimension by having each set of points be taken as a one dimensional list of $n$ lattice points and the ...
2
votes
1answer
272 views

2D inhomogeneous biharmonic equation with wedged edge

I'm solving bending of rectangular plate while, boundary conditions are I have found similar problem solved: datavoreconsulting.com/programming-tips/numerically-solving-pdes-mathematica-finite-...
3
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0answers
379 views

NDSolve`FiniteDifferenceDerivative how does it work? [closed]

I'm referring to Paritosh Mokhasi's blog post, where he uses, along with some other things, the...let's call it a function (?)... NDSolve`FiniteDifferenceDerivative. I was able to use it exactly the ...
2
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0answers
181 views

Solving nonlinear diffusion equation by NDSolve

I have checked all issues regarding to solving nonlinear PDE numerically. I however try to solve following equation via NDSolve function and get some troubles. And this is my code for above-...
1
vote
2answers
142 views

NDSolve problem

I am trying to solve numerically this differential equation s := NDSolve[{y''[x] + ω[x]*y[x] - 1/(y[x])^3 == 0, y[0] == 1, y'[0] == 0.3}, y, {x, 0, 10}] where <...
1
vote
1answer
154 views

Eigenvalues of 3D Laplacian on a spherical segment

To study the change in Laplacian eigenvalues on a spherical segment, I constructed a table of spherical segments using the code from here. I discretized the output using ...
3
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0answers
115 views

Coupled parabolic differential equations with time delay

Is it possible for NDSolve to solve delay partial differential equations with simple Neumann boundary conditions? An example I have is as below: ...
5
votes
1answer
505 views

Heat equation with nonlinear boundary condition involving time-derivative

The governing equation is shown as follows: I first try to employ the NDSolve, but it seems that Mathematica can not handle the fourth boundary condition. ...
7
votes
3answers
295 views

What is the best way to track the gradient to a local minimum?

Given a function of multiple variables, and some initial conditions, I would like an efficient way to track the gradient to the local minimum of that function. Two options spring to mind — to either ...
0
votes
0answers
265 views

Poisson equation with numerical right hand side

I run into the following problem. Linear Poisson equation works OK, eg, ...
4
votes
1answer
280 views

Laplace equation in a gapped rectangular domain with finite difference method

I have a situation that is shown by this picture: For this situation I have this code. ...
7
votes
1answer
575 views

Free Convective Heat Transfer of Non-Newtonian Power Law Fluids from a Vertical Plate

I am trying to solve a set of PDEs mentioned in this paper with NDSolve but facing quite a few issues. The PDE system is: ...
2
votes
0answers
319 views

Robin conditions using Finite Difference [closed]

I'm trying to solve the drift-diffusion equation $$\frac{\partial\rho}{\partial t} = \frac{\partial^2 \rho}{\partial x^2} + \frac{\partial \rho}{\partial x} \rho(x,t)$$ using a finite difference ...
0
votes
1answer
207 views

Why do I get this nonsensical result for NDSolve with a delay-differential-algebraic system?

I have the following delay-differential-algebraic system: \begin{align} c(t) =& \kappa\min (1, \tfrac{\beta}{\alpha} a(t)), \tag{1a}\\ a(t) =& \left\{ \begin{array}{lcl} \frac{\alpha}{\...
0
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0answers
271 views

FDM - Fluid Flow tutorial

I was reading a very nice post: http://blog.wolfram.com/2013/07/09/using-mathematica-to-simulate-and-visualize-fluid-flow-in-a-box/ But i am stuck for hours at the point where the author writes: <...
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0answers
152 views

Solving lengthy higher order pde using finite difference

I am attempting to solve a very massive pde (latex expression given below). I have done some work with finite difference before for relatively simple equations (like heat diffusion or the wave ...
7
votes
2answers
499 views

Updating a parameter each time step within a Finite Difference scheme?

So I am solving a PDE for a function $h(\theta,t)$ via finite difference scheme. The PDE has a function $Q$ in it, which I wish to update each time step depending on where $h$ lies. Firstly, let us ...
13
votes
2answers
885 views

Finite difference method not converging to correct steady state or conserving area?

I am working with the following PDE, which is an advection-diffusion type equation. It describes the movement of a fluid-fluid interface inside an annulus of inner radius $R_1$ and outer $R_2$ under ...
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vote
0answers
128 views

Speeding up solving nonlinear Schroedinger equation in 3D with NDSolve with periodic boundary conditions

I have a question on speeding up solving nonlinear Schroedinger equation in 3D with NDSolve with periodic boundary conditions. I build the ODE system with NDSolve...
5
votes
1answer
178 views

Kernel crashes when computing finite difference mixed derivative with respect to y & z but works fine when computing with respect to x & y or x & z?

Bug introduced in 8.0.4 or earlier, fixed in 11.0.0. I am using Mathematica 10.4.0 on Ubuntu 16.04. I am trying to solve a set of differential equations using finite difference method on an NxNxN ...
11
votes
1answer
494 views

NDSolve`FiniteDifferenceDerivative gives wrong result when the precision is not MachinePrecision

Bug introduced in 8 or earlier and persisting through 11.0.1 or later I want to get a pseudospectral differentiation matrix by NDSolve`FiniteDifferenceDerivative. ...
3
votes
1answer
303 views

Finite Difference - heat generation in a square (square within a square)

I am trying to develop a code that will allow me to solve the following problem. Lets say that I have a square block, inside that square block I have a smaller square box thats at a certain ...
2
votes
1answer
172 views

FiniteDifferenceDerivative of complex function in 2D--bug?

I want to compute partial derivatives of complex functions via finite difference approximation on two dimensional grid using NDsolve`FiniteDifferenceDerivative ...
1
vote
1answer
699 views

Partial differential equation, Finite difference Method

For my research project I am trying to solve following partial differential equation in Mathematica (V[r]*ψ[u, v])/4 + Derivative[1, 1][ψ][u, v] == 0 with ...
1
vote
0answers
329 views

How to derive finite-difference scheme automatically on a quite general stencil

Info This question is a generalization of the following one Derivation of numerical scheme for linear transport equation on a variable stencil. Statement of a problem Linear scalar hyperbolic (...
1
vote
1answer
196 views

Finite Difference equations - slow computations

The following code I have written runs very slow, in fact, so slow that it does not finish computation. ...
5
votes
1answer
637 views

How to vary finite difference approximation order in `NDSolve` at boundaries

According to here, NDSolve, when using the method of lines, creates partial derivatives in the spacial coordinate (lets talk just about one spacial coordinate for ...
1
vote
1answer
333 views

Poisson PDE in a rectangular domain

I found a solution for this problem, but this is in Scilab and I never use Scilab. Can anyone can help me to translate it in Mathematica? Here is the Link:http://imechanica.org/files/TorqueR.pdf I ...
2
votes
1answer
1k views

Laplace PDE in a polar coordinate system

I want to solve a Laplace PDE in a polar coordinate system with finite difference method, but I have a problem with boundary conditions at r = 0. Here is what I ...
5
votes
1answer
921 views

Very slow mathematica finite differences

I have a simple code which solves an equation by an explicit method (FTCS). It takes mathematica several minutes (mathematica 10.0.2) to finish the calculation while the same code in Fortran runs less ...
20
votes
2answers
1k views

Least effort to handle a point source inside the domain of PDE(s)

By point source I mean a constrained condition at one point inside the domain of PDE(s). For example: $$\frac{\partial ^2u(t,x,y)}{\partial t^2}=\frac{\partial ^2u(t,x,y)}{\partial x^2}+\frac{\...