Computational Science Stack Exchange
Computational Science Stack Exchange

Questions tagged [pde]

Partial differential equations (PDEs) are equations that relate the partial derivatives of a function of more than one variable. This tag is intended for questions on modeling phenomena with PDEs, solving PDEs, and other related aspects.

882 questions
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PDE solvers for Drift-diffusion and related models

I'm trying to simulate basic semiconductor models for pedagogical purposes--starting from the Drift-diffusion model. Although I don't want to use an off-the-shelf semiconductor simulator--I'll be learning other (common, recent or obscure) models, I…
Weaam
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3 answers

What is a scalable preconditioner for high-frequency Helmholtz?

Standard multigrid and domain decomposition methods do not work, but I have large 3D problems and direct solvers are not an option. What methods should I try? How are my choices affected by the following considerations? coefficients vary over…
Jed Brown
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discrete $L^p$ norms for non-uniform grid

I am reading a book on numerical methods and the square of the discrete $L^2$ norm is defined as $$||x||^2_2=h\sum_1^Nx^2_i$$ Every point gets a "weight", which is $h$, thus this is like an average over the squares of the values at all points. This…
Kamil
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8
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1 answer

Can I solve this time-independent PDE by adding a time derivative and marching in time?

I want to solve this PDE: Currently I have some code that will automatically generate pde solutions for a very similar pde that includes a time derivative (partial d/ partial t) using an ADI method. I'm wondering if there is a way to approximate…
phubaba
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5
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frozen coefficient vs. constant coefficient

This is a follow up to the question about the method of frozen coefficients. I think it deserves to be a separate question. The frozen coefficient problems are obtained by fixing the coefficients of the variable coefficient problem. For example,…
Kamil
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Solving diffusion PDE using finite differences

I need some hints on how to solve this diffusion equation ($\alpha, k_1,k_2$ and $k_3$ are constants): $$ {\partial P \over \partial y} + k_1 {\partial P \over \partial t} + \alpha P = {1 \over k_2} {1 \over x}{\partial \over \partial x} \left(x…
apm
5
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1 answer

Numerical method for solving PDE with non-linear boundary conditions

I have the following problem: \begin{align} \frac{\partial w}{\partial t} = \frac{1}{r^2} \frac{\partial}{\partial r}\left( r^2 D_1 \frac{\partial w}{\partial r} \right) \\ \\ \frac{dr_d}{dt} = C_1\frac{\left(C_2 + C_3 \left(…
maximus
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4
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2 answers

Reaction-diffusion system animation

I'm using following code to generate a picture of the reaction-diffusion process: Explicit Euler method too slow for reaction-diffusion problem I would like to obtain an animation effect. When I draw the solution step by step, it does give good…
Urank
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3 answers

Solving Laplace's equation on a domain with moving boundary

Consider a function $X(\xi,\nu)$, $2\pi$ periodic in $\xi$ satisfying $$\nabla^2 X = 0$$ in a domain $D$ with $\nabla = (\partial_{\xi},\partial_{\nu})$. If I know the values of $X$ on the boundary $\partial D$, what is the simplest (in a numerical…
Nick P
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3
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generating a non-uniform grid with Chebyshev discretization

I often see that it is common to put "more points" in the region of interest in the computational domain of the numerical method, i.e. use non-uniform grid. The proofs are usually done for the uniform grid though and then using some arguments like…
Kamil
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3
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2 answers

Discretizing time term for PDEs

I am actually confused about time stepping for PDEs. Before, I was distretizing time using backward Euler method for implicit formulation and I get a system of Algebraic equations to solve. Now during my research when I read papers, I find they …
MBM
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3
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conservative v non-conservative

I have always come across the terms "conservative" as opposed to "non-conservative" forms of equations in fluid mechanics. Is there a good reference that someone can share to clearly distinguish the two or shed some light on what they mean and what…
MSIngh
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3
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How to solve a nonlinear differential equation of this type

Trying to explain my problem better, I need to start with the diffusion Equation, which is the one that I am trying to solve. The Diffusion Equation \begin{equation} \dfrac{\partial u}{\partial t}=\dfrac 3 x\dfrac \partial{\partial x}\left(\sqrt x…
Nikko
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Numerics for heat equation

I want to simulate on my computer the solution of the heat equation in 3 space dimensions with Cauchy initial data, that is $$\partial_t u=Tr[A(x)\cdot \Delta u], u(0,x)=u_0(x) $$ where $u_0\in C(\mathbb{R}^3,\mathbb{R})$. Even if $A$ is constant…
john
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Pointwise convergence

I have seen a number of papers that propose a finite-difference method and then show the numerical results for it. Without providing a rigorous analysis(can be some summary or note or whatever, just no proofs involved) one can say method is of…
Kamil
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