# Questions tagged [parabolic-pde]

A partial differential equation that describes diffusion phenomena.

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### Is the Alternating-Directions Implicit method dependent on the space increment?

I am writing an Alternating-Directions Implicit Method for simple 2D diffusion ( \begin{equation*} \frac{df(x,y,t)}{dt}=D\Delta u \end{equation*}). Tridiagonal matrices are solved via Thomas ...
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### Numerical scheme for the heat equation on the icosahedral hexagonal grid

I have a predefined grid(like this) that is spawned from a regular icosahedradron. It consists of many hexagons and 12 pentagons (corresponding to icosahedradron vertexes). I can tweak the granularity ...
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### Nondimensionalization of a multi-component chemical diffusion equation

Edit I've modified the equations because they were wrong and I added the whole system, as asked by @Wolfgang I am trying to nondimensionalize a system of partial differential equations similar to 2nd ...
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### Is this a diffusion equation, or something else？

This is a time fractional PDE，where $0<\alpha<1$. I would like to know how to describe this equation and where does this equation come from.
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### "Optimal" domain partitioning in domain decomposition algorithms

When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
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### Derivation of a parabolic PDE using Alternating Direction Implicit method

I have a very simple question concerning Alternating Direction Implicit (ADI) Scheme. If I have an equation of the form: \begin{equation*} \frac{df(x,y,t)}{dt} = \nabla^2 f(x,y,t) + f(x,y,t) \end{...
1 vote
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### 3d schrodinger equation weak form

SCHRODINGER’S EQUATION $$-ih u_{t}(x,y,z,t) = \frac{h^2}{2m} u_{xx}(x,y,z,t)+ \frac{e^2}{r}u(x,y,z,t)$$ The potential $\frac{e^2}{r}$ is a variable coefficient. So, let’s take the free Schrodinger ... 42 views

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### 1D FEM for nonlinear diffusion coefficient

I want to solve with linear finite elements the equation $$\partial_t u = \partial_{x}(a(u)\partial_xu)$$ in the domain $t \in [0,1]$ and $x \in [-L,L]$. Here $a(u)$ is just a function of $u$. ...
1 vote
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### What will PDE discretization matrix look like for time and space? [duplicate]

Please note: this question is not a duplicate of this question since, while the PDE is the same, the nature of this question is different, i.e. the other question treats a different aspect of this PDE....
1 vote
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### Discretizing a parabolic PDE with finite volume method

I want to discretize the following parabolic PDE: $$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\ x\in\Omega \subset \mathbb{R}^2\\ \partial_n u = 0\\ u(t,0) = u_0(x)\ge 0, \alpha(x)>0$$ Given ...
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### Trying to compute the error from comparing two arrays

Some context: I am working with the Black-Scholes model.. I have an explicit (Black-Scholes) formula which is the exact solution to my problem. I have written code which implements a finite-difference ...
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### An interesting numerical pde problem

I'm somewhat struggling with how they are getting this scheme. This is a problem from Morton & Mayers book on numerical pde solutions. I think they are using a forward difference approximation for ...
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### Cauchy problem with a change of variables

Consider an advection-diffusion problem $$u_t+au_{xx}+bu_{x}=0,\quad x>0.$$ Now I want to remove the drift and rewrite the problem for the new domain. I use the change of variables $y=x-bt$, and ...
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### spurious oscillations Crank-Nicolson

I want to make sure I am clear with all the reasons why oscillations and how wild they get. I will summarize what I understood from the books with respect to the application of the Crank-Nicolson to ...
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I am a little confused about the truncation error and what it actually means. So, For example I have an explicit approximation to the heat equation $$\frac{u_{j}^{i+1}-u_{j}^{i}}{\delta t}=\frac{u_{j-... 0 votes 2 answers 1k views ### Help implementing finite difference scheme for heat equation I am trying to solve the following problem via a finite difference approximation: u_t = k \, u_{xx}, on 0 < x < L and t > 0; u(0,t) = u(L,t) = 0; u(x,0) = f(x). I take u(x,0) = ... 1 vote 0 answers 41 views ### explicit scheme stability restriction Looking at the plain heat equation u_t=u_{xx} the explicit scheme for it would look like the following iteration:$$u_{m,n+1}=\rho u_{m-1,n}+(1-2\rho)u_{m,n}+\rho u_{m+1,n}$$I noticed this equation ... 2 votes 1 answer 1k views ### heat equation on bounded and unbounded domain I have been reading about the heat equation and I am confused about uniqueness in the case when the domain is bounded and when it is not. In the book I am following, it is common to write the heat ... 3 votes 1 answer 140 views ### Space-time Galerkin of Burgers changes the convection speed tldr: Can space-time Galerkin schemes applied to convection-diffusion problems lead to effects on the convection velocity? For time t\in (0,1) and the spatial variable \xi \in (0,1), I am ... 1 vote 0 answers 78 views ### Stability of two PDEs I have the following two PDEs which I want to check for stability:$$u_t= u_{xx} , \ u(x,0)=1 , \ u_x(1,t)=-hu^4(1,t) , \ u_x(0,t) = 0 u_t = u_{xx}-\sin(x+t)+\cos(x+t) , \ u(x,0)=\cos(x) ,\ u_x(...
I want to solve the PDE equation numerically. For this, I started my study with something simple; heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial^2 x}$$ with the initial ...