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.
895
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How can I get more accurate electric scalar potential in 2D closed box?
I am trying to use poisson equation to plot the electric scalar potential in close 2D space. The details in in this video
and this one
The following in written in Matlab for quick prototype.
...
- 101
2
votes
1
answer
278
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Asking advice for implementation of Conservative Finite Difference Scheme for numerically solving Gross-Pitaevskii equation
I am trying to numerically solve the Gross-Pitaevskii equation for an impurity coupled with a one-dimensional weakly-interacting bosonic bath, given by (in dimensionless units):
\begin{align}
i \frac{\...
- 21
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0
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56
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What is the best finite volume method for the following equation?
I'm trying to create a partial differential equation that approximates 1-D climate in a rocky planet's atmosphere, which accounts for energy transport via radiation and convection. I am only ...
- 317
1
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0
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42
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How can I apply a mixed boundary condition to a multi-material heat transfer problem using Crank-Nicolson?
I am working on a mixed material model for a melting material and need to enforce both a Dirichlet and Neumann type condition at the interface. Subject to an external surface heat flux at the top of ...
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1
vote
1
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61
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How to constraint the tangential gradient on a boundary in FEniCS?
The problem I'm considering is a 2D scalar PDE.
The domain $\Omega$ is a disk with two holes $\partial\Omega_1$ and $\partial\Omega_2$ and an external boundary $\partial\Omega_0$.
The PDE and boundary ...
1
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1
answer
53
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How to obtain the transfer function between boundary condition and point of wave equation?
I am considering the wave equation with position varying material properties
$$
m(x) \frac{\partial^2 u}{\partial t^2} = \frac{\partial}{\partial x}\left(k(x) \frac{\partial u}{\partial x}\right), \...
- 470
1
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1
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201
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Numerically solving the Advection-diffusion equation with no-flux boundary condition leads to violation of mass conservation
I am trying to solve numerically the advection-diffusion equation of the following form
$$\frac{\partial C}{\partial t}=\alpha\frac{\partial^2 C}{\partial x^2}+\beta \frac{\partial C}{\partial x}$$
...
- 11
0
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46
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Vector poisson equation in cylindrical coordinates. What's wrong?
I am trying to solve this equation:
$$
\frac{\partial^2 A}{\partial r^2} + \frac{1}{r}\frac{\partial A}{\partial r} - \frac{A}{r^2} + \frac{\partial^2 A}{\partial z^2} = -g
$$
This is basically the ...
- 101
2
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0
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Which class of PDEs is well-suited for multistep methods?
I am experimenting with machine learning techniques to solve partial differential equations (PDEs). My goal is to use solutions from previous time steps to predict the solution at the next time step, ...
- 169
1
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1
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127
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How to refine $h$ and $\Delta t$ for convergence tests on evolution PDE
Setting
I am solving for $u(x,y,t)$ the wave equation $\partial_{tt} u - \partial_{xx} u = f$ on $(x,y)\in\Omega=[0,1]\times \mathbb{R}$ by splitting it into an equivalent first order system:
$$\...
- 13
1
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0
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120
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Iterative PDE solver for 1D Burgers equation
I am looking for an Iterative Numerical PDE solver for 1D Burgers equation. I need to have access to the intermediate solutions of the Numerical Solver. By iterative methods, I mean techniques which ...
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1
vote
1
answer
104
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Boundary conditions of a 2D explosion case
I want to solve this problem explosion test case. I was wondering about what are usual the boundary conditions for this type of problems. I want the wave to bounce of the boundary of the domain (let's ...
- 147
1
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2
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124
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Monotonicity of Errors with Respect to Step Sizes in Numerical Methods for PDEs
Consider a non-linear partial differential equation (PDE) (e.g., Burgers' equation) that is solved numerically using a finite difference method (or a similar approach). Suppose a grid search is ...
- 169
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1
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numerical schemes for 1D PDE: for smaller grid size there is an increased roundoff error, larger size more truncation, so sweet spot in between?
I had a discussion with a colleague today. He claimed that usually for a general numerical scheme for solving a general 1D PDE, for smaller grid size there is an increased roundoff error because of ...
- 445
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What exactly is a "unit-torus"?
I've seen references to the "unit torus" in papers such as this (Start of Sec 3.3, page 5). So, what exactly is a unit torus? Is it just a square or cube in d-dimensions with periodic ...
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How to implement boundary conditions for the Thomas algorithm
For my variable $U(t,x)$, I have implemented the thomas algorithm with $U_j^i$:
$$ a(x)U_{j-1}^{i+1}+ b(x)U_j^{i+1} + c(x)U_{j+1}^{i+1} = d(x)U_j^{i} $$
Then $\textbf{A}$ is a tridiagonal vector with ...
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0
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53
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Prof A. Stanoyevitch's finite difference based PDE matlab code
Where can one find Prof A. Stanoyevitch's finite difference based PDE matlab code? They have a book on such a topic but can't find the accompanying code.
Is it well received? It's not commonly talked ...
- 317
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1
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95
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Prof Lawrence Shampine's hpde matlab code
Where can one find Prof Lawrence Shampine's hpde matlab code?
Is it well received? It's not commonly talked about.
- 317
2
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0
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57
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Why the following discrete inequality are equal?
When reviewing papers on the numerical solutions of Partial Differential Equations (PDEs), I observed the following equation:
$$
(1-C\tau)||\theta^n||^2 \leq ||\theta^{n-1}||^2 + C\tau (h^{2r+2}),
$$
...
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1
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0
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84
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Seeking open-source PDE Solver for inhomogeneous material properties
I'm currently in search of an open-source PDE solver (Finite Element Method is preferred) that can effectively handle the challenge of material properties coefficients associated with each element in ...
- 11
2
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1
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304
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Literature request covering Chebyshev's pseudospectral collocation method
I would like to request some literature recommendations covering Chebyshev's pseudospectral collocation method for solving space-time PDEs. It would be nice if it even contained some example problems ...
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Solving AU = F using linalg.cg results in 0 iterations
I am working on solving the following PDE: $$\left(\mu_{x}\frac{\partial^{2}u}{\partial x^{2}}+\mu_{y}\frac{\partial^{2}u}{\partial y^{2}}\right)=f(x,y) \tag 1$$
Which is then discretised:
$$- \mu_{x} ...
- 43
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0
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109
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Solving a steady-state PDE
I'm working on trying to solve a steady state PDE in python and I am quite new to python and I am slightly confused on the implementation details. I have an example implementation of solving a non-...
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4
votes
1
answer
148
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Burger's equation (PDE) does not work with downwind difference?
I'm working on implementing the discretised Burger's equation. I am quite confused as to why it does not work when using a step-function and downwind difference formula. When using a step-function and ...
- 43
1
vote
1
answer
156
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Finite Difference method, ADI Scheme of Douglas and Rachford
I am trying to implement the ADI scheme of Douglas and Rachford.
For $p(X,Z,t)$, there is:
$$
\begin{gathered}
A=p^{n-1}+\Delta t_n\left[F_0\left(p^{n-1}, t_{n-1}\right)+F_1\left(p^{n-1}, t_{n-1}\...
- 11
4
votes
0
answers
126
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FVM for non-regular domain with triangular mesh
Setup
The 1D convection-diffusion equation is given by:
\begin{equation}\tag{1}
\frac{\partial u}{\partial t} + v \frac{\partial u}{\partial x} - \mu \frac{\partial^2 u}{\partial x^2} = 0,
\end{...
- 91
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0
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48
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Is a sort of "z-drift" the result of numerical precision errors in FDM?
Upon solving the 2D wave equation with Neumann boundary conditions $u_x = u_y = 0$ on a rectangular $10 \times 10 \times 10$ grid, I noticed something odd - $u$ seemed to shift upwards with time. This ...
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1
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0
answers
44
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Solution to the Liouville-Gibbs equation
What would be the approach to numerically solve for $\rho(x,t)$ the following equation with some initial conditions
$$\frac{\partial\rho}{\partial t}
+\sum_{i=1}^n\left(\frac{\partial(\rho g_i)}{\...
- 397
0
votes
1
answer
68
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Local truncation error of given implicit 1-step scheme
I'm given the 1-step implicit scheme $$y_{n+1} = y_n + \frac{h}{6}[4f(t_n, y_n) + 2f(t_{n+1}, y_{n+1}) + hf'(t_n, y_n)],$$
where $y'(t) = f(t, y)$, and I'm seeking the scheme's local truncation error. ...
- 1
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0
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64
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How to set Neumann BC for coupled transport problem in weak form?
Consider
$$\begin{aligned}
\partial_t v + b\cdot \nabla \phi &=0 \\
\partial_t \phi + b\cdot \nabla v &= 0
\end{aligned}$$
for $v:(x,y)\mapsto \mathbb{R}$, unknown and time-dependent ($...
3
votes
1
answer
176
views
How can we calculate mixed derivatives numerically using the Chebyshev derivative matrix?
Using the Chebyshev derivative matrix $D$, we can numerically approximate the first and second derivative of a function by doing matrix multiplication:
$${df(x) \over dx} = Df(x) \tag 1$$
$${d^2f(x) \...
1
vote
0
answers
103
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Method of lines for a mixed PDE
I am trying to solve the following PDE using the method of lines to discretize space, and then solve it as system of ODEs at each point in space using ODE15s:
subject to
and initial condition $w(z,t=...
- 11
1
vote
2
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196
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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 ...
- 197
0
votes
1
answer
80
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Determining the importance of different parameters in a simulation
Suppose that I have a function of, say, three parameters, $f(p_1,p_2,p_3)$ whose output is a field(s) (e.g. velocity field) and is dependent on some real-valued parameters (e.g. viscosity, density, ...
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1
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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 ...
- 197
2
votes
1
answer
99
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Symmetrization of Laplacian Matrix Operator (finite volumes)
The aim is to construct symmetric Laplacian matrix $L$ for 2D square domain.
I have the following discretization of second derivatives on non-uniform grid (will skip the steps of derivation, any ...
- 362
0
votes
1
answer
56
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Time discretisation after splitting a 4th order equation
Suppose we have a fourth-order parabolic PDE
$$\partial_t u + a(t)\Delta( \Delta u) - div( b(x,t)\nabla u) =0$$
We split the equation into two second-order equations by introducing $w = \Delta u$ thus
...
- 3
3
votes
2
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279
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Solving systems of advection-diffusion-reaction equations with finite element methods
I have been doing a lot of self-study on numerically solving PDEs so that I can solve system of linear and nonlinear Advection-Diffusion-Reaction (ADR) systems on complex meshes.
I have been watching ...
- 297
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0
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278
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Using solve_ivp for a PDE: how to handle multiple time-dependent variables?
I am trying to build a Python code that solves a set of coupled differential equations which will be spatially discretized by the method of lines advancing in time. I am planning to use ...
- 11
2
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0
answers
108
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How can we symbolically working out $\phi^4$ theory green's function/propagator and consequences in python?
I am having some difficulty calculating Green's function symbolically in Python for $\phi^4$ theory.
The specific rendition of the $\phi^4$ theory I have in mind can be written as follows.
$\mathcal{L}...
- 121
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0
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53
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PETSC: Solving a simpler PDE results in error while solving the original equation works in snes/tutorials/ex13.c
In snes/tutorials/ex13.c,
there is a function SetupPrimalProblem(),
which sets up the $f_0$ and $f_1$ in ...
- 101
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0
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81
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How to get damping matrix for structural model in FE analysis
I need to implement in C a method of obtaining transient solution of damped FE models based on modal results for a structural model (imported CAD geometry) defined with hysteretic (structural) damping....
- 1
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0
answers
43
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Verification of a Function Definition in Python
I want to write a function $f$ and it is defined as $f = - \nabla \cdot(|\nabla u|^{p-2} \nabla u) $ and I exact solution $u(x) = \tilde{u}(r) = 1 - \frac{p-1}{p-2} \left( s^{p/{p-1}} - (1-s)^{p/{p-1}}...
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votes
1
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Is using iterative methods to solve a linear system always superior to inversing the matrix?
I have a silly question. Is it always more computationally efficient to use iterative methods to solve for some matrix $A$, $Ax=b$, where $x$ and $b$ change but $A$ stays constant, compared to ...
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2
votes
2
answers
1k
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Problems solving 2D heat equation using physics-informed neural networks
I am trying to solve 2D heat equation using the physics-informed neural networks approach. The training loss is decreasing, but my final network outputs make no sense. I am using Python/Pytorch.
2D ...
2
votes
1
answer
118
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How to numerically solve differential equations involving sines, cosines and inverses of the unknown function?
I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method.
I find that the main idea is to ...
- 35
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0
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43
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A question from boundedness property of LMT
I have the following PDE
$$
u_t = k \Delta u + \alpha u H(u-c)
$$
I am trying to show the boundedness property to apply LMT. I get confused with the estimates of third term.
The space I have
$$V = \{...
- 143
3
votes
1
answer
189
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Numerical scheme for the level set equation that solves inverse mean curvature flow problems
I am considering the problem of simulating the evolution of an interface given as a curve in 2D (or surface in 3D) that evolves according to a velocity specified at the interface of the form:
$$\vec{v}...
- 33
3
votes
1
answer
189
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Discontinuous Galerkin for transport equation with non-constant advection
This question is mainly an inquiry about the usefulness of Discontinuous Galerkin (DG) for the time-independent transport equation of the form
$$\sigma u+\beta\cdot\nabla u =f,\;\;\;\text{on }\Omega\...
- 139
2
votes
1
answer
81
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Differential Equation with Forced Behavior
I'm attempting to solve a strange differential equation problem. My goal is to know if there are kinds of ODE solver packages to solve this kind of problem.
I'm solving a 1D Partial Differential ...
- 317