Questions tagged [finite-differences]

A method in numerical analysis which consists of approximating the derivatives of a solution of an ordinary or a partial differential equation. This leads to the solution of a linear system.

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Backward Differential Equation

There is a backward differential equation $$\frac{\partial}{\partial t} F(x,t) + \frac{\sigma^2}{2}\frac{\partial^2}{\partial x^2} F(x,t) = 0$$ for a function F(x,t). I see that the textbook rewrites ...
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Derivation of Product Rule for Finite Differences without Shift Operator

Let $h(x)=f(x)g(x)$, and let the $\Delta$ be the forward difference operator on functions be $$\Delta f := \frac{f_{j+1}-f_j}{\Delta x}$$ where $f_j = f(j\Delta x)$ Now I can derive the product rule ...
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Stable discretization of the heat equation using finite differences

Suppose $u$ is sufficiently smooth and satisfies \begin{align*} &u_t = u_{xx}, \quad 0<x<1, \quad t>0, \\ &u(x, 0) = f(x), \\ &u(0,t) = g(t), \\ &u(1,t) = h(t). \\ \end{...
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How can I verify the energy conservation of a finite difference scheme for Navier--Stokes equations in 2D?

How can I verify the energy conservation of a finite difference scheme for Navier--Stokes equations in 2D? I think that the answer with any finite difference scheme is enough to understand. Thanks in ...
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Advection-Diffusion Equation with two variables

Consider the following equation: $\frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial x^2} -u \frac{\partial T}{\partial x} -T \frac{\partial u}{\partial x}$ where $T$ is the ...
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Can the diffusive flux be represented for grids with non-constant thickness?

The figure below shows a water column that is separated into $3$ parts: Grid1, Grid2 and Grid3. Their thickness are $\operatorname{H1}$, $\operatorname{H2}$, and $\operatorname{H3}$. The concentration ...
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Existence and uniqueness of a finite difference approximation for a two-point boundary value problem

My question comes from Chapter 4 of "Partial Differential Equations with Numerical Methods" by Stig Larsson and Vidar Thomee. Consider the two-point boundary value problem \begin{equation} \...
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Finite difference with lexicographic ordering

2D convection diffusion equation with convection along x as:€(U xx + U yy)+aux = f(x,y) Where U xx and U yy are double partial derivatives w.r.t x and y. Ux is single derivative w.r.t x. a and € are ...
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How to limit oscillations of finite difference scheme at Neumann boundary conditions?

To give some background on the problem, I'm trying to solve the following PDE via finite differences: $$\frac{\partial\sigma}{\partial t}=\frac{\partial}{\partial x}\left[ \kappa(x,t)\left(\frac{\...
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Resources to study Nyström methods and their error analysis

I'm working on a problem that states that: The Nyström methods have the form: $$u_{n+r} = u_{n+r-2} + \Delta t \sum_{j=0}^{r}\beta_h f(u_{n+j}).$$ The problem is: Find the two step explicit and ...
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How to compute “One step error” for trapezoidal rule and two-step backward difference formulas

I've a homework problem that asks to compute "one step errors" (OSE for short) for the trapezoidal rule: ($k=$ step-size) $$y_{n+1} = y_n + \frac{k}{2}(f(t_n,y_n) + f(t_{n+1}, y_{n+1}))$$ ...
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Using finite difference for a simple second order BVP

I'm trying to solve this BVP using the finite difference method $$u_{xx}(x) +sin(x) = 0$$ with $u(0)=u(2\pi)=0$. When I solve it on python, I get the correct curve shape I expect, a sinusoidal one, ...
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Linearising Nonlinear Coupled Partial Differential Equations

Short: Is there any possible way to linearise the two equations: $$\partial_tf+\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$ $$\partial_tW+\partial_zW-\partial_zf=0,$$ where both f and W depend on ...
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Solve BVP $y''-3xy'-y=x$, $y(0)=y'(1)=0$ by power series, got differences to numerical solution

I was trying to solve the ODE $y''-3xy'-y=x$ by series. Well, we have regular points all over it so I thought it would be a nice shot, but I am having some problems finding the solutions. By proposing ...
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Smoothly connecting PDEs with finite differences

A PDE with non-smooth inhomogeneity Let $\mathcal{L}$ be a second-order, linear, elliptic differential operator acting on $\mathcal{C}^2([0,2]^2)$. I'm numerically solving the inhomogeneous PDE \begin{...
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Finite difference method on the Cauchy-Riemann PDEs

I made a forward fd-discretization of the Cauchy-Riemann PDEs but I am struggling to implement this in python. I have a quadratic mesh with heighτ = $2*\pi$. The dirichlet boundary conditions are at $...
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The last finite difference of $P$ considering in a general form

Problem: Consider a polynomial (monic) $P$ and finite differences as $Q(x):= P\left(x+\tfrac{1}{2n}\right)-P(x)$ and so on, for any real $a$. Show that the sum $$\sum_{k=1}^n (-1)^k \dbinom{n}{k} P\...
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Definition of error while computing order of convergence

I'm trying to verify order of convergence for implicit Euler method. Theory suggests that it should be $O(\Delta t + \Delta x^2) .$ We know the formula for order of convergence : $$\rho = \frac{log(...
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Deduce the order of convergence of explicit Euler

I'm reading the following lecture slides : https://courses.maths.ox.ac.uk/node/view_material/1191. See Page $86$ for my question. In the slides, they are verifying order of convergence of explicit ...
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Simulating PDE equation with Leapfrog scheme

I am trying to use Leapfrog scheme to solve the kdV equation: $u_{t} - 6 u u_{x} + u_{xxx} = 0$. The leapfrog scheme is used in this document: https://newtraell.cs.uchicago.edu/files/ms_paper/hmmorgan....
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Proof involving tridiagonal diagonally dominant matrix

$\mathbf{Background}$ Define a finite difference grid on the rectangle $[0,X] \times [0,T]$, $(x_j,t_n) = (x_0 + j\Delta x,n\Delta t) \text { for } 0 \leq j \leq M \text { and } 0 \leq n \leq N , \...
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Second-order accurate approximation to the first derivatives at the endpoints on a non-uniform mesh? (forward/backward finite difference)

I'm trying to derive a 2nd order forward approximation to a first derivative $u'(x)$ (and also $u''(x)$ is possible). I'm discretizing the first and second derivatives at the midpoints by $$u'(x_j) \...
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Stability Analysis Finite Difference Methods Black-Scholes PDE

$\mathbf{Background}$ Let $u = u(x,t)$ be the solution of the following forward parabolic initial boundary-value problem in one space dimension, $$u_t - a u_{xx} = f(x,t) \text { for } 0 < x < L ...
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Crank-Nicolson method for solving parabolic problem in 1D

Condiser the following initial-boundary value problem for $u = u(x,t),$ $$u_t - au_{xx} = f(x,t) \text { for } 0 < x< L \text { and } 0 < t < T,$$ along with boundary and initial ...
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stability of finite difference schemes for nonlinear pdes

I know that forward euler is probably not the most accurate approach for discretizing a PDE. However, due to its simplicity, it can be a good starting step. It is easy to code, however, it is also ...
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Impose PDE itself as Boundary Condition?

Consider, for example, the elliptic PDE $u_{x}+u_y+u_{xx}+u_{yy}=0$ for $(x,y)\in[0,\infty)^2$. Solution methods often require me to impose boundary conditions. Often, these arise naturally from ...
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How to solve this integral (and possibly Laplace transform) using a numerical iterative (finite difference) approach?

I am trying to implement a mathematical model for vibrational damping described in this article. They provide an equation for damping force ($F$) as a function of a spring constant $k$, a damping ...
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Introduction to finite differences

I'm looking into solving the wave equation for underwater sound as described in the Naval Underwater Systems Center's technical report Numerical Solutions of Underwater Acoustic Wave Propagation ...
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Chain rule in DARTS – Differentiable Architecture Search

For https://arxiv.org/pdf/1806.09055.pdf#page=4 , could anyone help to see how equation (7) is the chain rule result of equation (6) ?
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Fourth order finite difference

I am studying fourth order central finite difference (CFD) for space discretization of the Black Scholes PDE. I understood that the standard fourth order CFD for $N-1$ points is given by $$\frac{\...
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Handling a discontinuous grid in finite difference

I am supposed to model heat transfer through a cavity that absorbs concentrated solar radiation. The discretization of the cavity is sort of set by a previous ray-tracing model. I end up with ...
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Why is inter/extrapolation technique on finite difference table different than taylor's expansion of FDM?

I am studying finite difference methods on my free time. Finite Difference table So based on the table, if I want to extrapolate the next value of my function, I have to use ... $$f(x+dx)=f(x)+f'(x)+...
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How to solve this equation for the force of a piano hammer's felt in terms of finite differences?

The force from a piano hammer's felt on strings is typically modeled as a nonlinear spring as in $F(x) = k*(x^p)$ as per Hooke's Law. Additional terms may be added for hysteresis to describe the fact ...
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2answers
59 views

Nonlinear Implicit Finite Difference

How do you produce an implicit finite difference system with a nonlinear term in the pde? For example, if you have the reaction-diffusion equation: $$\frac{\partial{u}}{\partial{t}} = \Delta u + f(u)$$...
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Neumann Stability Analysis

$$ u_{j+1,m} = \frac{1}{2}\sigma(\sigma-1)u_{j,m+1}-(\sigma^1-1)u_{j,m}+\frac{1}{2}\sigma(\sigma+1)u_{j,m-1}\\ \sigma = \frac{c\Delta t}{\Delta x}\\ u_t+cu_x = 0 $$ I need find the Neumann Stability ...
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CFL Condition of general Finite Difference Scheme

I need to find the CFL Condition of the following Finite Difference Scheme. $$ u_t + cu_x = 0\\ u_{j+1,m} = Au_{j,m+3} + Bu_{j,m+2} + Cu_{j,m+1} + Du_{j,m} + Eu_{j,m-1} $$ So far I have the following ...
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Convergence of finite difference scheme for PDE $u_t+au_{xxx}=f$

From J.C. Strikwerda's book Finite Difference Schemes and Partial Differential Equations (SIAM), p. 57: 2.2.5. Show that the scheme $$ \frac{v_m^{n+1} - v_m^n}{k} + a\frac{v_{m+2}^{n}- 3 v_{m+1}^{n} +...
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Aproximate equation with Neumman's boundary conditions by using implicit finite difference method

I have this equation for $\rho$ $$ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x}(\rho u) = \varepsilon\frac{\partial^2 \rho}{\partial x^2},\quad 0\leq x\leq 1,\quad 0\leq t\leq T\\ \...
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What determines the stability and velocity of a 2D wave equation using finite difference method?

I have been able to master the 1D wave equation in finite differences, but I am struggling to get a similarly functioning 2D wave equation. For background I will review my understanding of the 1D ...
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A formula for operator $\frac D{e^D-1}$?

Is there a good unambiguous formula for the linear operator $\frac D{e^D-1}$? I mean, $$x^a\to B_a(x)$$ $$\ln x \to \psi(x)$$ $$e^x\to\frac{e^x}{e-1}$$ etc.
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What finite difference method do I use for below data?

The data in the table below represents the altitude $H$ (in feet) of a small rocket that is launched vertically upward as a function of time $t$ (in seconds): $$\begin{array}{c|cccccccc} t\text{ (s)}&...
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Difference quotient for higher derivatives

For a differentiable function $f:\mathbb R\to\mathbb R$, we have, by definition, $$f'(x)=\lim_{h\to0} \frac{f(x+h)-f(x)}h.$$ Is there a similar construction for higher derivatives, i.e. is it possible ...
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Backward Differential Equation with binomial tree

I'm trying to understand/solve the following question but I honestly don't know what it's even asking about. I've included my attempt following the picture of the question. I would approximate the ...
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Alternative ways of accuracy estimates for non-analytic cases for numerical purposes.

Let us start by example: consider a finite difference method $\mathcal{D}$ for a function $f$. If $f$ is analytic then one plugs the power expansion of $f$ into $\mathcal{D}$ and gets something like $$...
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Meaning of irregular error

In a paper I've read it is said that the error of the finite difference scheme is not regular since small changes in the exact solution can lead to big changes in the approximate solution. Can someone ...
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References for linear advection system with constant coefficients

I'm searching for references where I can found the study of finite difference schemes for linear advection system with constant coefficients. More specifically, I would like to read about: The Symbol, ...
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Why does Von Neumann Stability Analysis Work for Increasing functions?

Im having a hard time understanding why Von Neumann Stability Analysis is an appropriate measure of error propagation in numerical analysis. The condition of stability of a particular finite ...
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Difference equation corresponds to $y''=-y$

Convert the differential equation $y''=-y$ into a difference equation using the leapfrog method. My reference says: $$ \begin{bmatrix}1&0\\\Delta t&1\end{bmatrix}\begin{bmatrix}Y_{n+1}\\Y_{n+...
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How is this numerical step justified?

I'm following a course on numerical methods and we're discussing the finite-difference method in $2\text{D.}$ This question is about the proof that the system matrix is symmetric with dirichlet ...
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Numerically approximating the second order derivative at the end points of a closed interval

I have approximate the second order derivative of some function $f(x)$ on some interval $[a,b]$. Let $ h =\frac{b-a}{n}$ for some positive integer $n$, the discretization of $[a, b]$ will be the ...

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