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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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3
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0answers
87 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 ...
1
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0answers
166 views

3D Finite-Difference using Kronecker Products

As described in this Wikipedia article, a discrete laplacian matrix can be made for a 3D regular grid using Kronecker products. I'd like to use the same methodology for $(n-1)\times n$ matrices of the ...
1
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0answers
19 views

Does excitation type matter in a time-domain simulation/computation of a transfer function of an LTI system?

Let's say I am running a FDTD simulation of a wave-equation to determine a transfer function of an LTI system: \begin{equation} H(f) = \dfrac{Y(f)}{X(f)}\ \end{equation} where $Y(f)$ and $X(f)$ are ...
2
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1answer
101 views

interpolation 2D irregular nodes

Given a 2D irregular spaced data like shown in the figure, I would like to know how to find derivatives at '*' by interpolating the values at 'o'. Does lagrange 2D interpolation work at irregular ...
3
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4answers
1k views

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{\...
1
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1answer
281 views

(FD WENO) Correct characteristic decomposition of 2D Euler equations [closed]

After successful implementation of characteristic-wise finite-difference WENO method to 1D Euler equations, I'm moving to 2D equations on cartesian grid: $$ \frac{\partial U}{\partial t} + \frac{\...
0
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1answer
594 views

Finite Difference Method Limitations/Stability Criteria

Is it possible to solve an equation with only a single derivative such as: $$\frac{\partial U(x,t)}{\partial t} = A - BU(x,t)$$ with finite difference methods? I ask as I am trying to solve the ...
1
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2answers
133 views

How to compute WENO-reconstructed flux in local characteristic field?

I'm trying to apply finite-difference characteristic-wise WENO method to 1D Euler equations: $$ U_t + F(U)_x = 0, \; \;\text{where} \;\; U = (\rho, \rho u, e) \;\; \text{and} \;\;F(U) = (\rho u, \rho ...
1
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1answer
199 views

Does 1D component-wise Euler WENO work with shocks at all?

I'm trying to implement finite-difference WENO method to progressively complex equations and systems. At the moment I'm successful with singular scalar equations (constant advection and Burgers), and ...
1
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0answers
75 views

Computing the change of function at two close points without cancellation

I want to compute the difference $\Delta f(x_1,x_2) = f(x_1)-f(x_2)$ of a smooth function $f(x)$ at two points $x_1$ and $x_2$ which are close to each other. The magnitude of the expected result, $|\...
1
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0answers
145 views

Implementing Neumann boundary condition in nonlinear integro-differential equation

Problem I would like to solve a nonlinear integro-differential equations on a 2D square domain $\Omega$ subject to Neumann boundary conditions using finite differences: $$\frac{\partial u}{\partial t}...
1
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0answers
47 views

Extrapolating to non-fluid cells (for Shallow Water Equations), for a shore/beach?

The water height $h$ and 2d velocity field $(u,w)$ are "extrapolated to non-fluid-cells, i.e., setting $h$ equal to the value in the nearest fluid cell." [Bridson] I'm using finite differences. ...
1
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0answers
49 views

Discrepancy in estimating boundary stencil for finite difference method

I am trying to estimate the FD stencil for boundary as mentioned in this paper (section 4.1) using MATLAB. The stencil order (6th) is higher than the one mentioned in paper (4th). $$ f_1' +\alpha f_2'...
1
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1answer
383 views

How to implement finite difference method for one dimensional Navier-Stokes PDEs

I am trying to use backwards finite difference method to numerically solve a pair of partial differential equations: $\frac{\partial \left(pv\right)}{\partial x}+\frac{\partial p}{\partial t}=0$ $\...
3
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0answers
191 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} \...
1
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0answers
198 views

CFL condition of source term

Consider one dimensional hyperbolic pde $$u_t+f'(u)u_x=0$$ For the above problem ,CFL condition is $\Delta t\leq \dfrac{\Delta x}{|f'(u)|}.$ But if we include the source term $,S(u,t),$ which ...
1
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0answers
225 views

Crank-Nicolson scheme in space for advection equation

Consider the equation $$\frac{\partial}{\partial t}v(t,x)=\frac{\partial}{\partial x}v(t,x)$$, for $t,x\in\mathbb{R}$. I'd like to solve this equation forward in space and backward in time, ...
2
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4answers
751 views

Finite difference for mixed derivatives on nonuniform grid

I need to have a finite difference stencil for the mixed derivative $$f_{xy}$$ on nonuniform grids such as this one: Since I could not find a stencil in the literature, I tried to derive it by my ...
1
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0answers
85 views

How would you specify mixed boundary conditions for a 2D PDE in the matrix used for finite differences

I have the following PDE in 2D: $U_{x} + U_{xx} + U_y + U_{yy} + U_{xy} = f$ where $f$ is a constant. And I'm trying to create a matrix $A$ to solve the PDE through finite differences: $AU = f$. I ...
1
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2answers
105 views

First-order ODE scheme implementation giving less than first-order convergence?

I am solving the initial value problem $$ \frac{d}{dt} (E C_g) = -\delta, \quad E(0) = E_0, $$ for $E$, where $E$ and $C_g$ are functions of $t$, $C_g$ is completely known, and $\delta$ is a function ...
1
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0answers
72 views

Find B-spline coefficients from values on collocation points

The context of my question is how to compute high order derivates on direct numerical simulation of turbulent channel flow. It is of particular interest for fluid dynamics and turbulence research. I ...
3
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2answers
132 views

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 ...
1
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0answers
145 views

Time-dependent Schrodinger equation(time-dependent Hamiltonian)

This question was already asked here and this is a suitable form of the equation for numerically solving. \begin{align*} i\frac{\partial}{\partial t}u_{\ell}(r,t) = \Bigg(-\frac{1}{2} \frac{\partial ^...
0
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2answers
250 views

Finite difference methods for multidimensional coupled equations

My knowledge of finite difference is very basic so this could be very trivial. I've seen how multidimensional finite difference works for say fluid equations, but they are also dealing with a single ...
1
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0answers
71 views

Numerically solving generalized eigenproblem with Neumann conditions

I am interested in finding the eigenvalues/eigenfunctions of problems such as $$ \partial_{xx} u = \lambda \partial_{yy} u, $$ which can be solved as the generalised eigenvalue problem $$ \mathbf{A}...
1
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1answer
439 views

Crank-Nicolson method for inhomogeneous advection equation

Suppose we have the inhomogeneous advection equation $$\left(\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\right)u(t,x)=v(t,x)$$ for $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (...
6
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1answer
393 views

Correct eigenfunctions of Laplace operator by Finite Differences

I am trying to compute the eigenfunctions of the Laplace operator, i.e. finding $u$ in $$ -\nabla^2 u = \lambda u .$$ For now I am trying to do this in 1D, so $$ \nabla^2 = \partial_{xx} .$$ I am ...
1
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0answers
157 views

Accuracy of finite difference method for heat equation on a disk

To study an approximation for the heat equation $$\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2}=f(r,\theta)$$ on the ...
1
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1answer
143 views

Can this nonlinear advection-diffusion equation be discretized as to only have to solve SPD systems?

Consider a nonlinear advection-diffusion equation of the form $$ \frac{\partial u}{\partial t} = \nabla \cdot (a(u) \nabla b(u) - \vec{c}(u)u) \tag{1} $$ on a rectangular domain with Dirichlet ...
1
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1answer
153 views

Fully discrete finite element method for 1D dynamic euler-bernoulli beam problem

I am trying to solve a 1D initial boundary value problem in MATLAB using Finite Elements with time stepping, for the purpose of learning scientific computing and to build up to more difficult problems....
1
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1answer
287 views

Linearization in Finite Difference Method: Why?

I have a very basic question, and I hope some of you might be able to help me: In Fluid Dynamics, a common equation turning up time and time again is the so-called continuity equation: $$\frac{\...
1
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0answers
29 views

Simulation of a lens, insufficient points

I am simulating the propagation of a light pulse using the equation $$\frac{\partial}{\partial z}A=\frac{1}{2\cdot k_0}\nabla^2_rA$$ with $$k_0=\frac{2\pi}{\lambda_0}$$ The propagation with a step ...
4
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1answer
646 views

Von Neumann stability analysis with a constant term

I have a question concerning the von Neumann stability analysis of finite difference approximations of PDEs. There seem to be a wealth of online source explaining the application of this stability ...
0
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2answers
88 views

Parabolic differential equations with time delay

Let $d_1=1,d_2=2,a_{11}=\frac{5}{13},a_{12}=\frac{22}3,a_{21}=-2,a_{22}=\frac{6}7,\tau=\frac{5}7$, $\psi(t,x)=\cos^42x,\phi(t,x)=\frac{3}{13}x^4\sin^2 3x$, $\Omega=[0,200]$ How to solve: $$\left\{ \...
0
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1answer
388 views

Solving an equation in space and time using the Crank-Nicolson approach

Assume I have the following equation (light propagating in $z$-direction through the matter): $$id_zu+d^2_ru=0$$ with $u(z, r)$ being a complex wave. The time scale in this equation is $$t\equiv t_\...
2
votes
1answer
877 views

Solving an iterative, implicit Euler method in MATLAB

I'm trying to solve an iterative problem that includes an implicit (backwards) Euler method to find successive time values for a given function. The numerical problem is shown here: $$ \begin{...
1
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0answers
96 views

boundary condition in 2-D planar finite difference problem

I'm working on a 2-D planar finite difference code. My differencing scheme at the boundary nodes involves introducing ghost nodes in the computation. My code also involves a multi-dimensional ...
3
votes
2answers
529 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 ...
0
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1answer
159 views

Numerical solution of nonlinear thermoconductivity equation

I have to find and plot a numerical solution for the following equation (I have to write a solver): $$u_{t} = (u^2 u_x)_x$$ with the following conditions $u(0,t) =0, u(1,t) = \sqrt{\frac{2c-2}{t}}, u(...
6
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1answer
282 views

Gaussian Numerical Differentiation

Gaussian quadrature improves on Newton-Cotes formulas by allowing the abscissas to vary along with the weights in order to integrate higher order polynomials. Can this idea be extended to numerical ...
0
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2answers
130 views

Trying to compute the error from comparing two arrays

Some context: I am working with the Black-Scholes model.. I have an explicit (Black-Scholes) formula which is the exact solution to my problem. I have written code which implements a finite-difference ...
2
votes
2answers
592 views

MATLAB FFT Differentiation

I am trying to implement the Laplacian operator using Fourier Transform differentiation (https://en.wikibooks.org/wiki/Parallel_Spectral_Numerical_Methods/...
1
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0answers
63 views

Discretization of a multi-function term

I'm trying to do discretization to the following system: $\frac{{\partial \rho }}{{\partial t}} = - \frac{{\partial \rho }}{{\partial x}}u - \rho \frac{{\partial u}}{{\partial x}}$ $\frac{{\partial ...
1
vote
2answers
235 views

V-cycle Multigrid for 2D transient heat transfer on a square plate using finite difference

I'm currently developing a program to solve 2D transient state heat conduction on a square plate using the V-cycle multigrid. Althought my program is able to reach the steady state solution, it's ...
1
vote
0answers
155 views

Breather solutions of Sine-Gordon Using Finite Differences

I'm attempting to simulate a standing breather of the form $$ u(x,t)=4\tan^{-1}\left(\sqrt{3}\cos\left(\frac{t}{2}\right)sech\left(\frac{\sqrt{3}x}{2}\right)\right)$$ for the Sine-Gordon equation $$u_{...
1
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0answers
131 views

Smoothness indicator calculation for WENO methods

I am trying to calculate the smoothness indicators for the WENO methods using the method given by Jiang and Shu. $\beta_k = \sum_{l=1}^k \Delta x^{2l-1} \int_{x_{i-1/2}}^{x_{i+1/2}} \Big(\frac{\...
6
votes
1answer
204 views

Does mean removal increase accuracy of numerical differentiation?

I wish to compute the derivative of a vector through numerical differentiation. Let's say, we use a standard 2nd order central difference scheme, to arrive at a differentiation matrix, and apply it on ...
2
votes
2answers
392 views

Precision loss in Matrix-Vector product when applying Finite-Difference scheme

I am applying a 6th order Finite-Difference differentiation scheme as seen in http://www.scholarpedia.org/article/Method_of_lines/example_implementation/dss006 There seems to be severe numerical/...
7
votes
1answer
387 views

CFL condition in polar coordinates

In this question, I suggested that the Couran-Friedrichs-Lewy (CFL) condition for the wave equation in polar coordinates reads $$C = 2c\frac{\Delta t}{\Delta r \Delta \phi} \leq C_\max \enspace ,$$ ...
6
votes
3answers
357 views

ODE $x''(t)+\eta x'(t)+x(t)=0$ with the $\eta$ extremely small

I want to numerically solve the damped oscillator equation $$\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x=0 .$$ Usually it's very easy to solve this kind of ODE analytically and numerically. But now the ...