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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Iteratively solving 3D Poisson equation in MATLAB

I have written a function that sets up a sparse matrix A and RHS b for the 3D Poisson equation in a relatively efficient way. The set-up is nothing fancy: I have extended the 2D 5-point stencil to an ...
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Finite difference method (diffusion equation) for 3D spherical case

There is one 3D-diffusion IBVP for sphere (Dirichlet problem with zero conditions on the surface of sphere). The equation has the following view: $$\frac{\partial u}{\partial t}=\operatorname{div}\...
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Points on the interface

We consider the problem $\left\{\begin{matrix} k(x)\Delta u(x)=f(x) & \text{ in } \Omega\\ u=0 & \text{ in } \Gamma \end{matrix}\right.$ where $\Omega \subset \mathbb{R}^2$ open and ...
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Finite Difference Error Estimate for an Elliptic PDE with an Oscillatory Coefficient

Suppose I want to solve the PDE: $$-\nabla\cdot\left( a_\epsilon\left(x\right)\nabla u\right)=0 \text{ in } \Omega$$ $$u=g(x) \text{ on }\partial\Omega$$ Here, I assume that $a_\epsilon(x)=a(x,\frac{...
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Projection Method: Boundary condition on intermediate velocity field

I'm trying to solve variable density and viscosity Navier-Stokes equation using lagged pressure projection method. I'm solving for cavity problem as a test case now (once I get projection right, I ...
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Numerical solution of non-linear diffusion equation via finite-difference with the Crank-Nicolson method

I want to numerically solve the non-linear diffusion equation: $$ \frac{\partial}{\partial t} T(x,t)= \frac{\partial}{\partial x}\left(T^{5/2} \frac{\partial T}{\partial x} \right) $$ I want to use ...
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Implementing a finite difference method in Mathematica

I am trying to iterate the following equation $$ x_{k}(n+1)=x_k (n)-\epsilon (x_{k+1}(n)-2x_k(n) +x_{k-1}(n))+\sqrt{\epsilon}\; \eta_{k}(n) $$ where $n$ denotes which time step I'm on and $k$ is the ...
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Backwards Difference Implicit Method for Nonlinear Parabolic PDE in Python

Original Stack Overflow Question: https://stackoverflow.com/questions/65683788/indexerror-index-31-is-out-of-bounds-for-axis-1-with-size-31?noredirect=1#comment116218335_65683788 PDE: u_t = u_xx + u(...
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Accuracy of Explicit Euler method (finite difference) decreases as Δx decreases, shouldn't it increase?

The price of a commodity can be described by the Schwartz mean reverting SDE $$dS = \alpha(\mu-\log S)Sdt + \sigma S dW, \qquad \begin{array}.W = \text{ Standard Brownian motion} \\ \alpha = \text{ ...
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Kronecker product representation of the finite difference laplacian

The laplacian equation when discretized gives a system of linear equations that can then be solved. See the answer to this question: https://math.stackexchange.com/questions/3120948/discretization-...
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Matrix Representation of a Discretization for a Partial Differential Equation

I want to discretize the following problem \begin{cases} \mu \nabla^2u+(\lambda+\mu)\nabla \nabla\cdot u = \rho \frac{\partial^2u }{\partial t^2 } + \beta \frac{\partial u}{\partial t}\\ u(...
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Time discretization: Runge-Kutta methods vs. standard backward difference

I've recently written a code that solves the incompressible/low-Mach number formulation of the Navier-Stokes equation with high-order methods for both time and space. My advisor insisted that I use ...
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How could we solve coupled PDE with finite difference method and Newton-Raphson method?

I'm trying to solve coupled PDE by Crank-Nicolson (CN) and Newton-Raphson method with MATLAB. I have used CN method but not for coupled problem. Please if someone could help let me know to add more ...
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Comparison of Lattice Boltzmann Method vs Traditional Navier-Stokes based Methods

I have a choice of two options, analysing and implementing Lattice Boltzmann methods or traditional Navier Stokes based methods. I'm a CFD newbie and I have a rough idea (though not rigorous enough to ...
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Free open source C/C++ library to solve 2d Poisson equation using the finite difference method

I have been writing some code in C for particle-in-cell simulation. One of the steps of the PIC algorithm requires to solve (numerically) Poisson's equation $$ \Delta \varphi = - 4 \pi \rho. $$ ...
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4th order Padé scheme formula derivation

I am trying to derive the formula of the 4th order Padé scheme that passes through the points $x_i$, $x_{i-1}$ and $x_{i+1}$ $$\Big(\frac{\partial\phi}{\partial x} \Big)_i = -\frac{1}{4}\Big(\frac{\...
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2D Schrödinger time-independent finite difference and eigenvalues

I'm learning about numerical methods to obtain the eigenvalues of a system. I have to find the eigenvalues for the time-independent Schrödinger equation but I'm having some difficulties understanding ...
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transverse component for multidimensional advection in method of lines

So I inherited from some people a code that solves the advection-diffusion-reaction equation for a particular system. The original code was first implemented in 1D which worked fine in cartesian ...
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Numerically solve a PDE in Python with a term calculated by coarse-graining

I'm trying to solve a PDE in Python of the form, $\dfrac{\partial c(\mathbf{x}, t)}{\partial t} = \mathrm{D} \nabla^2 c(\mathbf{x}, t) -\gamma \rho(\mathbf{x}, t) c(\mathbf{x}, t)$ where $c$ ...
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Von Newman stability analysis for 2D acoustic wave equation explicit

Von Newman stability analysis for acoustic wave equation explicit centered differences: 2nd order time and space (N 2)'th order: \begin{eqnarray} U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} }{\...
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regularity of a solution and its affect on the global error

I am solving the following equation $u_t=x^2u_{xx}+\frac{x-y}{T-t}u_y$ with an initial data $u_0=max(y-C,0)$ for some $C$ in the domain of the numerical method. In time I am solving that on $[0,T]$. ...
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Defining electric current source excitations for surface integral equation formulations

In a finite difference (FD) based electromagnetic formulation based on a Yee cell grid, one can define electric current source excitations ($J$) on the $E$ field grid points. At a distance, the fields ...
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Stability of Crank-Nicholson for advection diffusion equation for spatial discretization other than finite differences second-order centered

Crank Nicholson is a time discretization method (see 4th equation here). From what I see around, you can use different space discretization, such as Finite elements. But for the linear advection-...
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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 ...
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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{\...
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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 ...
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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 ...
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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 }...
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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{\...
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Advection equation with finite difference: importance of forward, backward or centered difference formula for the first derivative

I am a newbie in finite difference methods, so I apologize in advance if the question is trivial. I am trying to solve the advenction equation, i.e. $\frac{\partial \phi(x,t)}{\partial t} + v \frac{\...
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735 views

Solving Ax = b with sparse A and sparse b

Let's suppose I'm numerically solving the Poisson equation for a delta function source: $$ \nabla^2 f(x) = \delta(x-x') $$ I can represent the Laplacian $\nabla^2$ using the finite difference method ...
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Why FVM can handle unstructured meshes while FDM cannot?

How come Finite Volume Method(FVM) handle the unstructured meshes and Finite difference Method cannot, whereas in FVM to approximate the fluxes at the boundary we use the central differencing? My ...
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Why do many people use FDM method to solve Stokes equations, i.e., saddle point matrix?

For numerical methods of the Stokes equations, with appropriate boundary: $$-\nabla^{2} \vec{u}+\nabla p=\overrightarrow{0}$$ $$\nabla \cdot \vec{u}=0$$ one may use FDM (finite difference method) ...
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CG question: is symmetry always necessary?

Consider the 1D Poisson equation $$\nabla^2 u = f.$$ Using finite difference method on cell corner data and a uniform grid with ghost points, I think we can write the system of equations with Neumann ...
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How do I program periodic boundary conditions? [duplicate]

Hi I have a code below that solves non linear coupled PDE's given Dirichlet boundary conditions. However I need to implement periodic boundary conditions. The periodic boundary conditions are ...
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Hyperbolic Equation PDE (Python)

I'm trying to solve the following first order hyperbolic PDE problem using method of lines: Hyperbolic Equation: $u_t = -u_x$ with initial condition: $u(0,x) = 0, 0 < x < 1$ boundary condition: ...
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1answer
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ODE boundary value problem relaxation method reference request

This is a somewhat basic question, I guess. Take the ODE boundary value problem $$ \frac1\lambda y''-y'=0, \qquad y(0) = 0, \quad y(1) = 1, $$ with the solution $$ y(x) = \frac{e^{x\lambda}-1}{e^\...
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How to obtain an implicit finite difference scheme for the wave equation?

Suppose I had the following problem: $U_{tt}=U_{xx}+U_{yy}$ in $\Omega=[0,1]\times[0,1]$ $U(x,y,0)=f(x,y)$ $U_{t}(x,y,0)=g(x,y)$ $U=0$ on $\partial \Omega$ I know that there is an explicit ...
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Finite difference method having a discontinuity

I am trying to understand the FDM which is a widely used method solving differential equations by using approximation below. $$\dfrac{\partial u}{\partial x}=\dfrac{u(i+1)-u(i-1)}{2\Delta x}$$ How can ...
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1answer
502 views

Finite difference methods in cylindrical and spherical co-ordinate systems

I am quite familiar with finite difference schemes in cartesian coordinates. The key point here is that every point in the cartesian grid is treated equally as the spacing between consecutive points ...
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227 views

Asymptotic error of forward Euler

I'm trying to understand the asymptotic error behaviour of forward Euler (finite difference method), as timesteps are decreased (refined), so I feel trust in the method of manufactured solutions (MMS)....
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1answer
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How can I solve wave equation for circular membrane in polar coordinates?

The original equation is $$\frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \...
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Effect of boundary condition on the local error

Any error analysis is based on the Taylor expansions. So, if I take a finite difference scheme, I can calculate the value of the function at any point using the known value at another node via Taylor. ...
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2answers
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Finite difference for a highly nonlinear equation - The wind within the forest

Based on the Navier-Stokes equations and a few parameterizations, the horizontal steady-state wind $u(z)$ within a forest of height H satisfies: $$ a\left(\frac{du}{dz}\right)^2 + b\frac{du}{dz} \...
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1answer
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Applying Neumann boundaries to Crank-Nicolson solution in python

Consider the heat equation $$u_t = \kappa u_{xx}$$ with boundary conditions of $$u(x,0)=0\\ u(0,t)=100\\ u(l,t)=0$$ Numerical analysis by pyton can be done with ...
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Finite Difference and Finite Volume as special cases of Finite Element

I have heard this many times that "FD is a special case of FE" and separately, that "FV is a special case of FE". However I have never come across a brief document that substantiates that claim, ...
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Why does multiplying two first derivative finite difference matrices not give the matrix for the second derivative?

The finite difference matrix for the first derivative is $\begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & -1 \end{bmatrix}$. The finite difference matrix for the second ...
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1answer
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Stable method for solving a HJB equation

I am considering finite difference methods and their error analysis for solving HJB equation of the following form: $$ v_t=|\sigma(x)v_x|,\quad x\in \mathbb{R}, $$ where $\sigma$ is a given function ...
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3answers
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Numerical approximation for a known exact solution of advection-dispersion equation

My goal is to create a numerical solution of 1D-solute transport (Convective-dispersion equation, CDE) to match it's analytical solution based on experimental data. The CDE can be written as (where, C=...
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1answer
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Generating a non-uniform grid

I am interested in generating a 1D non-uniform grid on the interval [0, L] with N points, where a region of width $\sigma$ and ...

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