# 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.

866 questions
Filter by
Sorted by
Tagged with
1 vote
150 views

### 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 ...
69 views

### 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, ...
135 views

### 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 ...
54 views

### 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 ...
53 views

### 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 ...
130 views

### 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 ...
1 vote
67 views

### 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 ...
73 views

980 views

### 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 ...
347 views

### 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 ...
111 views

### 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 ...
41 views

129 views

59 views

### Can I use MOL to solve 2D steady state PDE in terms of r and z spatial coordinates?

recently I need to solve a 2D steady state PDE equation. It’s not time dependent, and the only two independent variables are z and r direction. So far for this solution, I was thinking using Method ...
1 vote
30 views

### Applying spectral method to a damped, driven 2D bending-mode wave equation on an irregular domain with heterogeneous boundary conditions

I'm trying to model some two-dimensional waves, and am unsure how to combine my boundary conditions with spectral methods. The PDE I'd like to explore resembles the equations for damped, driven ...
155 views

### Can successive over-relaxation (SOR) method deal with ill-posed PDE BVP?

Recently, I've been struggling to understand the limitations and capabilities of the successive over-relaxation (SOR) method for boundary value problems which are ill-posed, such as, for instance, ...
170 views

1 vote
199 views

I recently started coding a small library of 2D PDE solvers (time dependent and time independent), and my first attempt was a 2D Poisson equation of the form: $$\nabla(\epsilon\nabla\varphi)=\nabla\... 4 votes 0 answers 220 views ### Spurious oscillations in solving diffusion problems using finite elements I've been struggling with this problem for a while so I hope someone can help me here. I'm trying to solve the McNabb-Foster equations for hydrogen diffusion in metals using a simple 1D finite element ... 1 vote 1 answer 99 views ### One dimensional C^1 finite elements I tried to solve a one dimensional biharmonic equation with finite elements. I wanted to use a conforming approach (as I simply do not know a lot about other approaches) and therefore was looking for ... 0 votes 1 answer 61 views ### Numerical solution of PDE with uniform initial condition I have a PDE like this$$ \frac{\partial h}{\partial t} = \bigg(\frac{\dot{L}}{L}\bigg)x\frac{\partial h}{\partial x} - \alpha\bigg[h^3\frac{\partial^3 h}{\partial x^3}\bigg] $$With boundary and ... 2 votes 1 answer 945 views ### How do you handle the singularity in polar or cylindrical coordinates? Governing equations in polar or cylindrical coordinates often have terms with \frac{1}{r} involved. At r = 0, such terms blow up to become a "singularity." The Cartesian version of such ... 3 votes 1 answer 146 views ### orthogonal basis functions on arbitrary domains and boundary conditions I'm interested in solving an inverse coefficient problem for a PDE. Let's say the field to be estimated is \theta. The conventional approach would be to use a finite element discretization for \... 4 votes 2 answers 150 views ### Numerical estimation of eigenfunctions of Laplacian Consider the Laplace equation,$$ \nabla^2 f(r,\theta,\phi) = 0 $$in spherical coordinates. We know that the solution to this equation can be derived analytically, and is given by,$$ f(r,\theta,\phi)...
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 ...