# 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.

820 questions
Filter by
Sorted by
Tagged with
36 views

• 235
1 vote
40 views

1 vote
24 views

32 views

### While solving Wave equation using FDM, how to solve and related eigenvalues

$\ t>0, x\in (0,\pi)$ \begin{cases} u_{tt}=u_{xx}\\ u(t,0) = u(t,\pi)=0\\ u(0,x)= exp(-32(x-(\pi/2))^2\\ u_t(0,x)=0 \end{cases} We assume that sufficient smoothness in $u$ and using \begin{cases} ...
• 695
30 views

### Application of boundary condition finite difference scheme

I am solving a version of the Laplace equation on a square ($a<x<b$, $0<y<h$) grid using finite differences. I have an analytical solution to my problem so I can easily check the ...
• 485
72 views

### Python code for Second Order ODE Initial Value problem using Finite Difference methods

I am trying to solve the following ODE \begin{aligned} & y^{\prime \prime}+2 y^{\prime}+y=0 \\ & y(0)=2 \quad y^{\prime}(0)=-1 \\ & 0 \leqslant x \leqslant 1 \end{aligned} Using ...
1 vote
49 views

### finite difference method for reaction diffusion equation

I am implementing a finite difference scheme for the following PDE: $$u_t=u_{xx}+f(u)$$ with the nonflux boundary condition and a given initial condition. The scheme is backward in time and central ...
• 157
39 views

### why can the result of the four point finite difference formula of a function, be greater than the exact derivative of the function?

if i have a function like $f(x) = e^{\sqrt{x+\frac{2}{x}}}$. why is the result of using the formula below, where $x$ is $2$ and $h$ is $0.1$, greater than the exact derivative of $f(x)$?
1 vote
28 views

17 views

### Constructing the matrices in finite difference method

I am trying to solve the following PDE using finite difference: \begin{aligned} \frac{\partial}{\partial t} u(x, t)-\frac{\partial^2}{\partial x^2} u(x, t) & =x, \quad x \in(0,1), \quad t \in(0,1),...
52 views

• 389
36 views

### Convergence of Finite Difference Scheme for Helmholtz equation

Consider the 2-dimensional Helmholtz equation $$-\Delta u-k^2 u= f.$$ I'm trying to proof convergence of the standard finite difference scheme (using the 5-point-stencil for the Laplacian). I have ...
1 vote
38 views

### Spectrum of discretized Laplace operator with homogeneous Neumann B.C.s

Consider the Laplace operator over a 1D domain with homogeneous Neumann B.C.s. The cell-centered Finite Volumes discretization of this operator on a uniform grid looks as follows, which is a well-...
• 61
61 views

• 581
72 views

### Did I get Godunov's scheme right?

I want to implement Godunov's scheme in order to simulate the nonlinear LWR-type equation $$\partial_t u + \partial_x (u(1-u)) = 0, \quad u(0, \cdot) = u_0.$$ The update step is ($n$ denotes time ...
80 views

### Boundary Value Problems Solving with Central Difference

I have differential equation $u''(x)+9u(x) = \cos(2x), \; x \in [0, π/2]$ the boundary conditions are $u(0)=1$, $u(\pi / 2) = -1$, with $h = 0.2$ Now I replaced $u''(x)$ with centered difference ...
34 views

### Curve Fitting and Parameter Estimation Study

I have a problem with an experimental data analysis to obtain unknown parameters. The experimental data can be described by the equation: $y_k-y_{k-1}=\eta_k (x_k^{1-\alpha_k}-x_{k-1}^{1-\alpha_k})$ ...
48 views

### Numerical Partial Differential Equation

The solution of the difference diagram for some partial differential equations from the Fourier transform and Fourier analysis can be written in ${U_{n}}^{k}=q^{k}e^{in\xi}$ form. The condition for ...
1 vote
339 views

• 11
51 views

### The real CFL condition for cylindrical laplacian

I've been exploring the CFL (Courant-Friedrichs-Lewy) condition in polar coordinates and have observed that previous inquiries haven't yielded a satisfactory answer. I've come across this paper which ...
57 views

### Transform Lagrangian with a square root

I have an action given by, $$S = \int^{\tau_f}_{\tau_0} d\tau \left(\frac{1}{z^3}\sqrt{-f(u,z) \dot{u}^2 + 2 \dot{u} \dot{z} + \dot{x}^2} + \frac{2 \dot{z}}{z^3} \right),$$ ...
• 485
49 views

### Coupled non-linear PDE analytical solution to verify numerical solution.

as part of a computational module we were given the following coupled PDEs and solved them numerically using finite difference methods: I got the following graphs representing the numerical solution ...
• 103
Assuming I have a differential equation $$\frac{\partial u}{\partial t} = \frac{1}{f(x,t)} F(u,x,t)$$ The Crank-Nicolson scheme would have the equation discretized as such  \Big[ \frac{\partial u}{...