Questions tagged [finite-difference]
Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.
195
questions with no upvoted or accepted answers
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73 views
2nd-order TVD criteria for flux-limiter
Consider a nonlinear hyperbolic conservation equation:
$$
\partial_{t}u = -\partial_{x}f(u)
$$
The spatial derivative of $f(u)$ may approximated after a spatial discretization by $x_{j}=j\Delta x$
$$
\...
5
votes
0answers
91 views
Evolutionary dynamics in vascularised tumors, PDE-ODE coupled system
I have to solve the following PDE-ODE system
$$ \displaystyle{\partial_{t} n = \bigl[a(s) - b(s)(y - h(s))^{2} - d\int_{\mathbb{R}} n \, dy \; + \; \beta \, \partial^2_{yy} n \;}
\\\\
\...
5
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0answers
84 views
Numerical methods for the continuity equation with Sobolev vector field
Consider the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$.
...
5
votes
0answers
222 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 ...
5
votes
0answers
244 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 ...
5
votes
0answers
1k 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^{-...
5
votes
0answers
547 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 ...
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170 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 ...
5
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0answers
794 views
Can Gauss-Seidel/SOR (preconditioned?) be applied to a non-diagonally dominant matrix?
After applying finite difference method to a Laplace/Poisson problem always arises a diagonal dominant system of equations that can be solved with Gauss-Seidel or SOR methods. If the original PDE does ...
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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 ...
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118 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
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80 views
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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0answers
60 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 ...
4
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0answers
224 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:
$$
\frac{\partial^2 \omega_n}{\partial r^2}+\frac{1}{r}\frac{\...
4
votes
0answers
1k views
How to implement boundary conditions on Finite Difference WENO5 scheme for the Euler equations
I'm implementing a Finite Difference WENO5 with Lax-Friedrich flux splitting on a uniform, structured grid to solve the 2D Euler equations of fluid dynamics on a rectangular domain in cartesian ...
4
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117 views
The numerical solution of a (very ugly) set of integro-diferential equations
I've been having fun playing around with a simple model for reaction in a co-flow fluidized bed (I'm afraid I'm a chemical engineer, not a computer scientist). Unfortunately, simple physical ...
4
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1k views
2D wave equation with Mur boundary condition - setting up the RHS and solving (time-steps)
I am trying to solve a 2D wave equation implicitly using FD with central approximations with the following boundary conditions
$$\begin{align}
&u=2\sin\left(\frac{2\pi}{5}t\right)\quad \text{at }...
4
votes
0answers
149 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{\...
3
votes
0answers
71 views
TVD Lax-Wendroff with non-constant velocity
I am dealing with a linear advection equation with a non-constant velocity, where I would like to apply a TVD Lax-Wendroff scheme in 1D.
The equation is the following:
\begin{equation}
\frac{\...
3
votes
0answers
69 views
Solving PDEs: What is the best way to deal with non-banded/dense jacobians?
I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
3
votes
0answers
64 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
59 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 ...
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94 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
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90 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
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58 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
89 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
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117 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
39 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
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167 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
75 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
109 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
511 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
97 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
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251 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
481 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
339 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
293 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
479 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
215 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
320 views
Efficient assembly of finite element matrix(coupled equations case)
I noticed this post, where spalloc and sparse are recommended for efficient assembly in Matlab. I personally use sparse ...
3
votes
0answers
352 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
63 views
Divergence on wave equation simulation
I'm currenly working on my own PDE solver for non-linear simulations in python. I've done succesfully simulations for KdV and Fisher's equation, but now I'm playing with second order derivatives in ...
2
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0answers
70 views
Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion
I am trying to solve the following coupled partial differential equations with a finite difference scheme:
$$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$
$$\partial_tW+v\partial_zW-\...
2
votes
0answers
43 views
How to solve this boundary value problem which has more unknown than equation on MATLAB
I need your helps about solving the problem below with MATLAB. I am trying to solve 2D Stress Wave Propagation problem by using FDTD(Finite difference time domain) method on the cylindrical coord. I ...
2
votes
0answers
42 views
In sights into why higher order finite differencing method leads faster to instability
I was playing around with numerically solving the 1D wave equation with density and stiffness varying with position using central differencing methods and noticed that for certain discretization steps ...
2
votes
0answers
59 views
Haw to apply central difference to viscous flux in energy equation?
In many modern papers Navier-Stokes equations are solved with finite-difference or finite-volume methods using WENO reconstruction for non-viscous fluxes and central differences for viscous ones. It ...
2
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62 views
How does the "Stable Fluids" algorithm by Jos Stam relate to the SIMPLE and PISO algorithms?
The "Stable Fluids" paper (*) by Jos Stam starts by acknowledging that "Our method cannot be found in the computational fluids literature, since it is custom made for computer graphics applications. ...
2
votes
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75 views
How to determine the finite difference coefficient matrix in 2D with periodic BC?
I'm solving a PDE in matlab using ode15s, and since the spatial dimension is 2, and number of variables grow large very quickly, I need to supply the structure of ...
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35 views
Convergent Finite Difference Scheme for Parabolic Equation
Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$
where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$.
In Strikwerda's book, the ADI scehme \begin{align*}
\left( 1 - \frac{...
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votes
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57 views
Numerical errors due to terms of the form $\frac{1}{r}$ (r goes to 0 at the boundary) while using finite difference method
I am trying to solve a system of differential equations using finite difference method.
There are few terms of the form $\frac{A(r)}{r}$, both $A(r)$ and r go to zero at the boundary. Analytically ...
