# 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.

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### Numerical methods for the continuity equation with Sobolev vector field

Consider the continuity equation $$\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,$$ with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
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### Discrete sine and cosine transform for mixed derivatives

Using sine and cosine transforms to solve Poisson's equation with Dirichlet boundary conditions seem quite standard nowadays (see, e.g., here or Table 2 in this paper). In the case of Poisson's ...
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### Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - \...
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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 ...
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### Algorithm for face based data-structure - CFD

Good morning I'm trying to develop an unstructured CFD code to solve Euler equations in a finite-volume (cell-centered) context (learning purposes). I was able to build from a cgns file some basic ...
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### Eigenvalues of a Laplacian operator on an irregular mesh

I have the following setting: An irregularly-shaped domain, expressed as a mesh of points A Laplacian operator, together with boundary conditions I am looking for the eigenvalues of that operator, i....
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### Phase dislocations and numerical accuracy

I am solving the nonlinear Schrodinger equation (NLSE), $$A_t+iA_{xx}+i|A|^2A=0$$ where $A$ is a complex valued function, which can be written as $A=ae^{i\theta}$ for $a,\theta$ real. Now, for ...
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### Form of nonlinear diffusion equation

Consider the following nonlinear diffusion problem, $$\frac{\partial u}{\partial t} = x^{-2}\frac{\partial}{\partial x}\left(x^2 u^4 \frac{\partial u}{\partial x}\right), \quad 0 < x < 1$$ We ...
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### Numerically solving a 6th order non-linear differential equation in Matlab

I've posted yesterday a question about solving a non linear equation : it was not clear so I am reformulating my question. I am trying to solve a high-order non linear differential equation presented ...
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### Solving PDEs: What is the best way to deal with non-banded/dense jacobians?

I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
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### What are the practical differences between a Galerkin finite element method vs an orthogonal collocation method?

What are the practical differences between a Galerkin finite element method vs an orthogonal collocation method? The best reference I've found thus far as been a paper titled The Performance of the ...
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### Spectral/hp-finite elements for 4th order PDEs

Does anyone know of references discussing the solution of 4th order PDEs by way of spectral/hp-finite element methods? Specifically, I'm interested in the extension of the spectral element method, ...
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### How should I choose the knot sequence when using B-splines as a basis for solving a PDE?

I'm looking to solve the Schrödinger equation with a basis made of a tensor product of basis splines. A number of papers describe calculations made with a program designed this way, but they never ...
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### Centered finite volume scheme for an advective term on an unstructured/irregular/non-uniform grid

Consider the continuity equation $$\frac{\partial u}{\partial t} + \frac{\partial \Phi}{\partial x} = 0$$ $$\Phi = au + b\frac{\partial u}{\partial x}$$ Suppose I want to solve the above using ...
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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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### solve Ax=b for outrigger A matrix python

I implement Crank-Nicolson 2D finite-difference method. I get a matrix A which is banded with 1 band above and below the main diagonal, but also contains 2 additional bands , further apart from the ...
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### Divergence on wave equation simulation

I'm currenly working on my own PDE solver for non-linear simulations in python. I've done succesfully simulations for KdV and Fisher's equation, but now I'm playing with second order derivatives in ...
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### In sights into why higher order finite differencing method leads faster to instability

I was playing around with numerically solving the 1D wave equation with density and stiffness varying with position using central differencing methods and noticed that for certain discretization steps ...
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I am trying to solve a set of linear PDEs of the form $$F\left(\vec{y},\frac{\partial \vec{y}}{\partial x},\frac{\partial^2 \vec{y}}{\partial x^2},\frac{\partial \vec{y}}{\partial t}\right)=0.$$ To ...