# 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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### Algorithmically finding mixed-derivative coefficients using finite differences

Suppose $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is a differentiable (for simplicity, let's assume ${C}^\infty$) function, and I would like to find the following partial derivatives numerically (at ...
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### First Order Central Divided Difference

Show that the first-order central divided difference is given by: $$f'(x_i) = \frac{f(X_{i+1}) - f(X_{i-1})}{X_{i+1}-X_{i-1}} + O(h^2)$$ Note: Please explain in the simplest way possible, I'm still ...
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### Finding the CFL condition of second order $u_j^{n+1} =u_j^n +aH(u_{j-1}^n)+bH(u_j^n)+cH(u_{j+1}^n)$ with $u_t=H(u)_{xx}$ and $0\le H'(u)\le d$.

We have the following partial differential equation $$u_t =H(u)_{xx},~~~ 0\le x<1$$ with an initial condition $u(x,0) = f(x)$ and periodic boundary condition. Here $0 \le H'(u) \le d$. Consider the ...
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### How will the solution of Viscous Burgers’ equation (with zero diffusion) behave as we keep increasing time?

The viscous Burgers’ equation without diffusion is given by: $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$$ Now the solution should satisfy the implicit form: $u=f(x-ut)$. However ...
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### Numerical Stability of Non-Linear, Autonomous, PDEs on a Lie Algebra

Background: The dynamics of my system are modeled by a system of PDE's. I adopt some finite difference approximations to simplify into a system of ODE's and want to study its numerical stability to ...
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### Crank Nicolson method error analysis

I am trying to solve the diffusion PDE: $$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$$ using the CN discretization. I have implemented the method in Matlab, quiet ...
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### Solving a 2nd Order ODE using Finite Difference Method when Mixed Boundary Conditions are given

The problem I'm looking at is $$y'' + 3.05 y' -2.85 = 0$$ with the boundary conditions $y(0) = 1$ and $y'(1) = 0.0305$. After obtaining the algebraic set of equations using FDM, I'm not sure how the ...
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I understand how to write the heat equation: $\frac{\partial u}{dt}=c\frac{\partial ^2u}{\partial x^2}$ in numerical finite difference form implicitly (see wiki): $\frac{u_{j}^{n+1} - u_{j}^{n}}{k} =c\... 3 votes 1 answer 88 views ### Truncation error, finite differences Consider the following FDM problem: Find$u$such that $$-u^{\prime \prime}(x)+b(x) u^{\prime}(x)+c(x) u(x)=f(x) ~~\text { in }(0,1),$$ and conditions$u(0) = u(1) = 0, where $$b(x)=x^{2}, \qquad ... 0 votes 1 answer 94 views ### Can the left side of the equation be truncated to the first term? So, I am deriving a high-order compact finite difference scheme, and got into the equation below: \begin{multline} \delta t^2 \{1 + \dfrac{h^2}{12} (\delta x^2 + \delta y^2 + \delta z^2) + ... 0 votes 0 answers 44 views ### Finite Difference approximation of third mixed partial derivative I am looking for a Finite Difference approximation of third mixed partial derivative, specifically: \frac{\partial^3 u}{\partial x \partial y \partial z} (and I guess that will be an O(h^3) ... 3 votes 1 answer 130 views ### Finite differences and Neumann/Periodic BC of a fourth order PDE Before I start, I have already read this similar thread and didn't get anywhere unfortunately. I have a PDE of the form$$ \frac{\partial u}{\partial t} = \nabla^2[2u(u-1)(2u-1) - \gamma \nabla^2 u] ... 0 votes 0 answers 31 views ### Finite difference method boundary value problem Here is a boundary problem \begin{align} -\partial_x^2 \tilde{u}(x) + q(x)\tilde{u}(x) & = f(x), \; x \in (0,1)\\ \partial_x \tilde{u}(0) &= \alpha \\ \tilde{u}(1) &= \beta \end{align} ... 0 votes 0 answers 14 views ### Discretization of second order nonlinear ODE using finite difference with pseudo-spectral method I have the following equation that was discretized using pseudo-spectral method in 2D with periodic BCs that works. \nabla_{\perp} \cdot \left( n \nabla_{\perp} \psi \right) = \textbf{S} $$... 0 votes 0 answers 41 views ### What is the relation between the finite difference method's solution and the weak solution? I never understood the justification of some authors using the Finite Difference Method to solve a PDE numerically while theoretically they only proved the existence of weak solutions. Is the use of ... 0 votes 0 answers 18 views ### Mixed boundary conditions in a finite difference 2PDE I have a 2nd order PDE wave equation for a grid of points in space and time:$$\frac{\partial^2u}{\partial t^2}=a^2\frac{\partial^2u}{\partial x^2}$$I need to make a model according to finite ... 1 vote 0 answers 53 views ### Index reduction of a DAE from a PDE system I have a system of 2 non-linear, coupled PDEs that I would like to transform to a stiff ODE system to solve them using the method of lines. The 2 equations:$$\begin{align} \frac{\partial \phi}{\... 1 vote 1 answer 43 views ### Finite difference and recurence relation I am trying to find a solution to the recurrence relation $$g_{k}(n) = n g_{k-1}(n) - n g_{k-1} (n-1)$$ such thatg_k(n)$is written as a summation over$g_0(n)$for various$n$. Attempt I started ... 2 votes 0 answers 59 views ### Discrete Dirichlet problem has a unique solution I am studying Artin's Algebra. In Chapter 1 Exercise M11, he asks to show that every discrete Dirichlet problem on a finite discrete set in plane has a unique solution. This is essentially the ... 0 votes 0 answers 14 views ### Numerical scheme for non-linear PDE I have a function$V(t,x,y)$, with$t\in [0, T]$denoting the time,$x,y\in \mathbb{R}$. The function$V(t, x, y)$satisfies the following PDE: \begin{equation}\label{eq:hjb2} \partial_t V - a \Big(\... 0 votes 0 answers 27 views ### What is a correct way to compute the discrete derivative of a difference equation? Consider a discrete map$x(t+1) = f(x(t)), \quad\forall t\in \mathbb{Z}$For example,$f(x(t)) = (x(t))^3$. From searching Wikipedia on forward difference, the discrete equilvalent of the derivative ... 0 votes 0 answers 18 views ### Aproximating a function for a Robin/Fourier and Dirichlet condition First of all, im sorry if it seems silly but I haven't been able to come up with an answer: Lets say I'm trying to solve for$P(x)$with$x \in (0,L)$having$N$points in the domain using finite ... 0 votes 0 answers 25 views ### How do you convert negative integer powers to falling powers? Is there a systematic way to convert$x^{-n}$to falling powers for positive$n$? i.e. something like Stirling numbers, but that works when the exponent is negative. 0 votes 1 answer 81 views ### Solve the ode using fixed-point iteration I have the following equation$-[(1+u^{4})u_x]_x = sin(x)+sin(5x)$, where the domain is$[0,2\pi]u(0)=u(2\pi)=0$for boundaries. How to find a numerical solution for$u\$ using numerical method? My ...
I am going over stability for numerical PDEs and am trying to differentiate between the "regular" stability criteria for IVPs: $$||\textbf{u}^{n+1} ||\leq K ||\textbf{u}^0||$$ and the von ...