# 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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### ON the Kronecker product form of the laplacian matrix

It's well known that if we use 2nd order, centered, finite differences for the Laplace operator, we have that the matrix can be written as $$K=I \otimes A + A \otimes I$$ where $I$ is the identity ...
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### How do you handle the singularity in polar or cylindrical coordinates?

Governing equations in polar or cylindrical coordinates often have terms with $\frac{1}{r}$ involved. At $r = 0$, such terms blow up to become a "singularity." The Cartesian version of such ...
1 vote
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### finding discretization error in Burger equation

I was reading the paper given in the link http://www.unige.ch/~hairer/preprints/parareal.pdf and I have a problem in understanding in page 10 for the Burger equation on implementing Parareal method ...
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### How can I check mass conservation when solving the advection equation using an upwind scheme?

My question is how to keep track of the "mass" being advected out of a model domain, for the 1-D advection equation, and an upwind differencing scheme. Following is the background Consider ...
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### Dense factorization specialized for RBF-FD method

In RBF-FD methods (see Fornberg & Flyer. A Primer on Radial Basis Functions with Application to the Geosciences. SIAM, 2015. Chapter 5.), the finite-difference stencil coefficients for a set of ...
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### How to define a stretched coordinate perfectly matched layer (PML) parameters for maximum absorbtion?

I have written a MATLAB code for solving Maxwell's equations (electromagnetic wave propagation) in 3D with perfectly matched layer (PML) boundaries. I am using a stretched coordinate PML, but I see ...
1 vote
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### Calculating the species mass consumption from implicit reaction-term in diffusion-reaction equation

The 1D diffusion equation with a chemical source term has the following form: $$\frac{\partial Y}{\partial t} = D \frac{\partial^2 Y}{\partial x^2} - k Y,$$ where $Y$ is the molar concentration of the ...
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### 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 ...
1 vote
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### How does Tannehill impose boundary conditions when coding the Parabolized Navier Stokes on an Implicit Finite Differences Scheme?

I'm trying to implement the scheme he describes on his book "Computational Fluid Mechanics and Heat Transfer" on Chap.9 and I'm having trouble imposing BC. I don’t get how he imposes them. I ...
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### Closed (Robin) boundaries in advection-diffusion equation with FDM

I am solving the equation $$\frac{\partial \phi}{\partial t} = \frac{\partial}{\partial x} \left( D \frac{\partial \phi}{\partial x} + v\phi \right)$$ using finite differences. I want to include ...
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### How to accelerate the computing of implicit finite difference method for heat conduction between two solids

Edit on May 3rd: I have found the problem. Because the difference of between $k_1$ and $k_2$ is huge, a very small time step need to be chosen so that the right green part can "feel" the ...
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### Eigenvectors of Laplacian

I am studying introduction to Multigrid methods. In all tutorials, authors write that eigenvectors of Laplacian (1D, finite difference) are given as $w_k(x_i) = \sin(k \pi x_i),$ where $x_i$ is a ...
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### How can I correctly determine velocity of a point inside a grid after using mixed finite element method to solve Poisson equation?

I am using the mixed finite element method (MFEM) to solve the Poisson equation: $$\Delta h = 0,$$where $h$ denotes hydraulic pressure. The MFEM could determine the normal flux rate, $q_n$, through ...
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### Mineral dissolution and solute transport around a solid

I am trying to simulate solute transport of acid (HCl) and consequent mineral dissolution around a grain (calcite). The governing equation for transport is the advection-diffusion equation, given as: ...
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### Shock Capturing Methods for Shallow Water Equations

I am looking for some help finding a numerical solution to the shallow water equations: $\partial_tu+\partial_x(u^2/2+g\eta)=0$ $\partial_t \eta+\partial_x(u\eta)=0$. where $u$ is the depth averaged ...
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### Oscillating eigenvectors for 2d-laplace operator

I am trying to calculate the eigenvectors of a Laplace-operator in 2d, with boundary conditions equal to $u=0$ if $x, y$ are outside of a rectangle defined as $(0.5, 0.5), (1.5, 1.5)$. For that I used ...
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### solve Ax=b for outrigger A matrix python

I implement Crank-Nicolson 2D finite-difference method. I get a matrix A which is banded with 1 band above and below the main diagonal, but also contains 2 additional bands , further apart from the ...
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### Requesting for Finite Difference Methods reference in Portuguese or English

Crossposted on Mathematics SE I have been assigned a group project for an introductory Linear Algebra subject on Finite Difference Methods and sparse matrices. Our professor advised we use Gilbert ...
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### Transparent Boundary Conditions relationship with intermediate BCs in ADI-PR method

I have read some materials about ADI - PR method with the aim to understand how to put boundary conditions in my 2D scheme which solves the Time-Dependent Schrodinger Equation. All the theory I read ...
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### 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 ...
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### Comparison on adaptive mesh refinement on finite elements and finite differences

My current work requires using (Adaptive Mesh Refinement) AMR to resolve multi scale physics. I have a general question whether finite element is better than finite difference in this aspect or not. I ...
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### Finite-difference software for solving custom equations

Are there any good, easy to use, software for simulating the evolution of systems of generic differential equations? I know there are custom programs for various specific circumstances (such as ...
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### Weighted Jacobi Not Working on 1D Poisson (Issue with Optimal $\omega$)

I've been trying to learn some numerical linear algebra, and I decided to try to implement the weighted Jacobi method to the 1D Poisson problem $$-u''(x)=f(x),\qquad u(0)=a,\ u(1)=b,$$ where we ...
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### Lattice Boltzmann methods vs Navier stokes/ other eulerian methods for *water* simulation

Note, there is already a question here, however the answers don't answer the original question, let alone specific considerations when dealing with nearly in-compressible fluids (water). Another ...
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### Need help in attempting to solve an MHD eigenvalue problem

Background I am attempting to numerically solve the ideal MHD equations in normal mode form for a Harris current sheet. The linearized perturbed MHD equations can be written in a normal mode form:  -...
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### What does it mean for a finite difference method to be conditionally stable? Specifically when solving the diffusion equation

The diffusion equation is ∂u/∂t =∂^2u/∂x^2. Consider using an explicit finite difference method to solve it. What does it mean for that explicit finite difference method to be conditionally stable?
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### How do I identify negative group speeds?

This question is a continuation of one of my other questions. I've been trying to show that collocated (non-staggered) grids can suffer from negative group speeds in the linearized shallow water ...
I have a set of coupled pdes which I want to solve using finite-difference, of which one is nonlinear. The three linear pdes for quantities $T_f$, $T_s$ and $c$ are convection-diffusion-reaction-like ...