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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41 views

Stability plot of upward difference implicit time

I am analyzing the stability of 1D convection (advection) equation as shown in the picture. When I derive the equations as shown I want to get rid of the complex number. I`m asking if those stability ...
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1answer
93 views

Discretization of a non-linear ODE using FDM isn't grid indepenent

I am trying to solve the ODE : $\frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2$ + using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ...
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35 views

discretization of advection diffusion with variable coefficients

I am looking for help to find a somewhat stable FD numerical scheme for the advection diffusion equation posed on a curve $(x(r),y(r))$. The equation becomes $$u_t=\alpha(r) u_r +\beta(r) u_{rr}+f(r,t)...
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43 views

High Running Time and Suboptimal Accuracy of 2D Wave Equation Solver with Finite Differences

Im trying to solve the following 2D wave equation: $$u_{tt} = u_{xx} + u_{yy}, \hspace{3mm} u(x,y,0) = \cos(4 \pi x) \sin(4 \pi y), \hspace{3mm} u_t(x,y,0) = 0$$ with the periodic boundary condition ...
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11 views

(FD WENO) Correct symmetry boundary condition for Euler equations

I'm trying to solve 2D Euler equations in axisymmetric formulation with finite-difference WENO scheme. I found some info on high-order boundary conditions for plane formulation (in this thesis, for ...
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1answer
51 views

2 point BVP solver: how to compute errors

Background I am working with chapter 2 in LeVeque's book: https://faculty.washington.edu/rjl/fdmbook/ I have build my own solver in Python to solve the 2 point BVP: $$ \epsilon u''+u(u'-1) =0 , \\ u(0)...
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101 views

Trouble Implementing 1d Wave Equation Finite Difference Solver

Im trying to solve the 1d Wave Equation on $x \in \mathbb{R}, t > 0$: $$u_{tt} = c^2u_{xx}, \hspace{5mm} u(x,0) = \cos(4 \pi x), \hspace{5mm} u_t(x,0) = 0$$ with $c = 1$ and a periodic boundary ...
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55 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 ...
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1answer
73 views

Incorporating heat flux into Laplace Equation

I need to find the temperature distribution of a square plate using the Laplace equation by using FDM: $$ \frac{d^2T}{dx^2} + \frac{d^2T}{dy^2} = 0$$ But there is a heat flux entering from the top ...
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36 views

Results blow up when number of intervals is increases (Yee algorithm FDTD, dielectric sphere)

I have been trying to write a program that analyses EM wave scattering by a dielectric sphere for a project. The reference is Sadiku's book Numerical Methods in electromagnetics Edition 3. Now the ...
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66 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-\...
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41 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 ...
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27 views

velocity in CFL condition

I am studying the evolution of the density and velocity field of a core in a molecular cloud in 1 D. I defined the radial grid (let us say x between 100 and 101) and the time grid. I am using the ...
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28 views

Numerical dispersion in FDTD

I was reading the book "Computational Electrodynamics: The FDTD method" by Taflove and Hagness, probably the most cited book when it comes to the FDTD method in Electromagnetics. In the ...
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1answer
60 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 ...
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82 views

Why does FDTD and FIT disregard Gauss's law?

This is a reformulation of a question I asked a couple of days ago. I'm posting it again because I believe the previous post was very unclear, I will probably delete the previous question. My question ...
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86 views

Verification of order of convergence of Implicit Euler Method to numerically solve Black-Scholes PDE

I'm trying to verify the order of convergence for implicit Euler method to numerically solve Black-Scholes PDE. Theory says that it should be $O(\Delta t + \Delta S^2).$ My code is working absolutely ...
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34 views

When to stop iterations in SOR solver for 3D Poisson equation

I'm writing a solver (in C) for 3D incompressible fluids, using the finite-differences method, and I'm finding a somewhat surprising behaviour: the solver provides "good-looking" solutions, ...
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1answer
58 views

Implementing Dirichlet BC for the Advection-Diffusion equation using a second-order Upwind Scheme finite difference discretization

i am implementing a Matlab code to solve the following equation numerically : $$ (\frac{\partial c}{\partial t} =-D_{e} \frac{\partial^2 c}{\partial z^2} +U_{z}\frac{\partial c}{\partial z}) $$ with ...
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56 views

“This DAE appears to be of index greater than 1” daeic12 (line76) error code

Hi I am trying to solve a set of pde converted into ODE and DAE using central finite difference method. I have used the MATLAB 'solve' command to determine the coefficients of fictitious nodes for ...
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1answer
75 views

Compute 1st derivative with backward difference approximation in python

I am trying to write a function to compute 1st derivative with backward difference approximation. $ u'(x_i) = \frac{u(x_i) - u(x_i - \Delta_x)}{\Delta x} \equiv D_- u(x_i).$ And for the first point, ...
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68 views

Technique to find the CFL condition using the Galerkin method in space and finite-difference in time?

I am using the Galerkin method (Discontinuous to be precise) to discretize in space the scalar linear wave equation and the explicit second order centered finite difference scheme to discretize in ...
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72 views

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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43 views

Integrating a wavelike equation with absorbing boundary conditions

I am trying to numerically solve the following equation: $\frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}+V(x) \phi(x, t)=0$ On some domain, with: $\phi(x, 0) = I(x)$ ...
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148 views

Which finite difference better approximates $uu'$?

I want to approximate $uu'$ with a finite difference. On the one hand, it seems to be $$(uu')_i=u_i\frac{u_{i+1}-u_{i-1}}{2\Delta t}=\frac{u_iu_{i+1}-u_iu_{i-1}}{2\Delta t}$$ On the other hand, $$(uu')...
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67 views

Numerical solution of the advection equation with Crank–Nicolson finite difference method

I need to implement a numerical scheme for the solution of the one-dimensional advection equation $$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$ $$ \\ C(x,t) = \...
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38 views

Devising Convergent Numerical Scheme for PDE

I'm currently looking at the PDE \begin{align*} u_t + \left[x(1-y) - (1-x)\right]u_x - (1-y) u_y + (z-xy) u_z = (z-xy) u_{xy} - (1-x)u& \\ \end{align*} with \begin{align*} u(x,y,z,0) = 1& \\ ...
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1answer
65 views

Fix for FD WENO method for multi-component compressible flows

I'm solving two-dimensional four-component compressible Navier-Stokes equations with finite-difference WENO approach. The equations are pretty standard: $$ \frac{\partial U}{\partial t} + \frac{\...
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62 views

How to make a directed graph symmetric?

Say I have a directed graph given as an adjacency matrix $A$ in CSR format represented by the arrays ia (row indexes) and ja (...
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2answers
126 views

How to apply central difference to viscous fluxes in 2D Navier-Stokes equations?

I'm trying to solve 2D unsteady compressible Navier-Stokes equations with finite-difference or finite-volume method. Here is the system, it's pretty standard: $$ \frac{\partial U}{\partial t} + \frac{...
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1answer
73 views

Finite Difference for Advection Equation With Source

I'm trying to find a convergent finite difference scheme for the PDE \begin{equation} \begin{split} u_t + (x-1) u_x &= (x-1)u, \hspace{.5cm} x \in [0,1] \\ u(x,0) &= 1 \\ u(1,t) &= 1. \\ \...
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92 views

Solving 1D wave equation with finite difference method

I've written a code in Python to solve the 1D wave equation with the finite difference method (the explicit and the implicit methods). I'm trying to perform a mesh convergence study to estimate the ...
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40 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 ...
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49 views

Finding a CFD paper with extra degree of freedom variable in mass conservation

I am trying to find a paper that I saw about a year ago. I am not sure of the actual date of the paper. I believe it was a finite difference CFD paper. The interesting part of the paper was the ...
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1answer
49 views

How I can derive the Neuman boundary condition of this system of hyperbolic equations in 1D?

I would like to research the Neuman boundary that can verify the following problem $\begin{aligned} &\text { (} P \text { )}\left\{\begin{array}{l} \frac{\partial U}{\partial t}(x, t)+A \frac{\...
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50 views

discretizing advection equation with variable wave speed + stability

I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form ...
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1answer
111 views

(Lack of) Availability of Finite-Difference library for simple 2D PDEs

I would like to solve two types of simple 2D problems, namely the stationary heat equation on an L shaped geometry like this: And also compute the magnetostactic field in an air gap of the following ...
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50 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 ...
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66 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 $x_{j}=j\Delta x$ $$ \...
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1answer
124 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) = ...
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118 views

Simplest way to “upgrade” from Euler equations to Navier-Stokes equations in FV or FD framework

I have quite a lot of experience solving unsteady Euler equations, including multi-component ones, with in house-coded finite-difference and finite-volume methods, including MacCormack and MUSCLE ...
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1answer
91 views

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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1answer
73 views

How is the mixed 2nd partial derivative simplified to a more efficient form?

I'm implementing the Finite Different 2nd and 3rd derivatives in my research and naturally I'm looking for the most efficient approach. From https://en.wikipedia.org/wiki/Finite_difference#cite_ref-...
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1answer
76 views

If analytically, a function is not differentiable at a point, does it make sense to write a finite difference code for the function at that point?

What would happen if I wrote a finite-difference code evaluated at a point where the function isn't differentiable analytically? I'm trying to think analytically vs numerically. Thanks,
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108 views

Solution of Cahn-Hilliard equation

I need to solve the Cahn-Hilliard equation $$\frac{\partial u}{\partial t} = \Delta(f(u) - \epsilon^2\Delta u), \hspace{.5cm}(x, t)\in \Omega\times(0, T],$$ using mixed formulation \begin{equation}\...
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1answer
145 views

3D laplacian operator

I have been unable to find the equivalent of the 5-point stencil finite differences for the Laplacian operator. In 2 dimensions for me it is clear that, using the finite difference method: $$ \nabla_{...
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1answer
64 views

Parity for artificial dissipation term in a finite-difference solution

I have a doubt regarding the signal of the dissipative term in a finite difference solution for an equation of the form $$ \frac{\partial u}{\partial x}+f(x)=0, u(0)=0 $$ In which $f$ is an odd ...
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50 views

numerical solution to pde on an ellipse

Looking for advice on discretization (preferably finite difference) schemes for pdes on curves in general, but in this case it is an ellipse (so given by $(a\cos(r), b\sin(r)$). The problem is the ...
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2answers
227 views

No flux Neumann boundary condition for non-stationary PDE equivalent to Dirichlet boundary?

When using no flux Neumann boundary conditions (i.e. zero derivative to the normal on the boundary) in a non-stationary PDE, I don't seem to recognize the difference to using Dirichlet boundary ...
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1answer
99 views

Method of Lines: How to simplify Jacobian with periodic BCs?

Consider the advection equation $$\frac{\partial u}{\partial t}+c(x)\frac{\partial u}{\partial x}=0.$$ With periodic boundary condtitions in $x$ with period $L$, i.e. $u(x,t)=u(x+L,t)$ and initial ...

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