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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Does anyone know how to add a forcing term at the center of a cicular membrane?
I am here once again searching for wisdom, as some of you might notice I asked a related question a few days ago. And now I am struggling to add a forcing term at the center of the membrane, in order ...
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Can this finite difference dispersion be eliminated somehow?
I am trying to solve the wave equation
$$ {\partial ^2u(t,x) \over \partial x^2} = {\partial ^2u(t,x) \over \partial t^2} \tag1 $$
with the following boundary and initial conditions:
$$ {\partial u \...
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Solving the wave equation for a circular membrane in polar cordinates
As you see this mode is not right, unless for what i understand
And the initial conditions were
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First-order modified Patankar–Euler scheme (MPE)
Is the first-order Modified Patankar–Euler scheme (MPE) an implicit or explicit method?
Is there an open-source code implementing the MPE scheme for a system of ODEs?
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Discretization of generalized kinetic term in 2D Poisson partial differential equation
A typical 2D Poisson PDE is given as
$$\nabla^2\varphi(x, y)=f(x, y)$$
where the Laplacian term, $\nabla^2\varphi$, can to some degree be interpreted as the kinetic energy (given proper scaling) (...
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Perfect Conducting (PC) and Perfect Insulating (PI) boundary conditions for magnetic induction in MHD problems in cylindrical coordinates
I understand the computational implementation of the PC and PI bcs in cartesian case. In terms of the magnetic field components for PI, we have $B_x = B_y = B_z = 0$, and for PC, we have $\partial_x ...
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Modeling contamination diffusion in a draining container, part 2
Part 1, but I'll repeat here. This time we'll move the top boundary.
I have a container of water with initial volume $V_0$ and height $L$, open at the top which receives a constant mass flux $\phi$. A ...
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Closed formula to diagonalize discretized (perhaps randomized) Laplacians
I was wondering whether there is a closed formula for the eigenvalues and eigenvectors of the discretized Laplacian in (edit) $[0,1]^n$ with a uniform grid, using what I imagine is a $2n+1$ stencil.
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Automatic Differentiation In the Presence of Jump Points
I have a complex monte-carlo cashflow model that traditionally uses the finite difference (FD) method to calculate its derivative at any given point. To improve model performance, I coded forward-mode ...
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Modeling contamination diffusion in a draining container
I asked this on the Physics site, but it fits much better here, and I now can ask a more specific question.
For the scope of the problem, I have a 250-mL bottle filled with pure water and a sampling ...
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Can any discretization scheme reproduce the kane quasi-linear dispersion relation?
It is straightforward to solve the eigenproblem $$\hat{k}^2 \psi = E\psi$$using a finite-difference method. But what about$$\hat{k}^2 \psi = E(1+\alpha E)\psi$$I know it is possible to first solve the ...
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First derivative central differences with reflecting boundary conditions
I have the following problem in 1-D:
\begin{equation}
\partial_t u(t,x) = v(x)\partial_x u(t,x) + \kappa(x) \partial_{xx} u(t,x),\,x\in\Omega;\quad \partial_{\nu}u(t,x) = 0, \,x\in\partial\Omega.
\end{...
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Stability for the 2d diffusion equation
I'm reading this set of notes on numerical stability of the diffusion equation. I can't seem to derive equation (7.17) from the previous equation. I've tried using $\sin^2(\theta)=\frac{1-\cos(2\theta)...
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How and when to impose inhomogeneous Dirichlet boundary conditions in Newton method solver for a PDE?
I am applying a central Finite Difference scheme in space and an implicit Euler scheme in time on a variant of the 2d Burgers equation, of the form:$$u_t + uu_x + uu_y = \nu(u_{xx} + u_{yy})$$ where $...
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Literature request for pinning the corner singularities in finite differences
Corner singularities cause instability of finite difference solutions. They occur at the intersection of boundary conditions. I was recently going through an online video course that was explaining ...