# 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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### What are criteria to choose between finite-differences and finite-elements

I am used to thinking of finite-differences as a special case of finite-elements, on a very constrained grid. So what are the conditions on how to choose between Finite Difference Method (FDM) and ...
2answers
4k views

### Strange oscillation when solving the advection equation by finite-difference with fully closed Neumann boundary conditions (reflection at boundaries)

I am trying to solving the advection equation but have a strange oscillation appearing in the solution when the wave reflects from the boundaries. If anybody has seen this artefact before I would be ...
4answers
2k views

### Why is local conservation important when solving PDEs?

Engineers often insist on using locally conservative methods such as finite volume, conservative finite difference, or discontinuous Galerkin methods for solving PDEs. What can go wrong when using a ...
2answers
4k views

### Is Crank-Nicolson a stable discretization scheme for Reaction-Diffusion-Advection (convection) equation?

I am not very familiar with the common discretization schemes for PDEs. I know that Crank-Nicolson is popular scheme for discretizing the diffusion equation. Is also a good choice for the advection ...
3answers
4k views

### What is the purpose of using integration by parts in deriving a weak form for FEM discretization?

When going from the strong form of a PDE to the FEM form it seems one should always do this by first stating the variational form. To do this you multiply the strong form by an element in some (...
3answers
943 views

### Why is the time dimension special?

Generally speaking, I've heard numerical analysts utter the opinion that "Of course, mathematically speaking, time is just another dimension, but still, time is special" How to justify this? In ...
1answer
5k views

### Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation

I don't understand the different behaviour of the advection-diffusion equation when I apply different boundary conditions. My motivation is the simulation of a real physical quantity (particle density)...
1answer
11k views

### Why is Newton's method not converging?

I am using PETSc's nonlinear solver package SNES to solve a system of nonlinear equations obtained by discretizing a partial differential equation. How can I determine why the solver is not converging ...
8answers
3k views

1answer
688 views

### Are there any open source inverse-based multilevel ILU implementations?

I am very impressed with the serial performance of multilevel inverse-based ILU preconditioners, particularly for heterogeneous Helmholtz, but I am surprised to not be able to find any open source ...
3answers
641 views

### PDEs in Many Dimensions

I know that most methods of finding approximate solutions to PDEs scale poorly with the number of dimensions, and that Monte Carlo is used for situations that call for ~100 dimensions. What are good ...
3answers
925 views

### Is variable scaling essential when solving some PDE problems numerically?

In semiconductor simulation, it is common that the equations are scaled so they have normalised values. For example, in extreme cases electron density in semiconductors can vary over 18 order of ...
3answers
746 views

### 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 ...
4answers
6k views

### Boundary conditions for the advection equation discretized by a finite difference method

I am trying to find some resources to help explain how to choose boundary conditions when using finite difference methods to solve PDEs. The books and notes which I currently have access to all say ...
5answers
526 views

### Examples of PDE computations using parallelism in both space and time

In the numerical solution of initial boundary value PDEs, it is very common to employ parallelism in space. It is much less common to employ some form of parallelism in the time discretization, and ...
1answer
1k views

### What are the relative benefits of using Adams-Moulton over Adams-Bashforth algorithm?

I am solving a system of two coupled PDE's in two spatial dimensions and in time computationally. Since the function evaluations are expensive, I would like to use a multistep method (initialised ...
3answers
6k views

1answer
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### What are possible methods to solve compressible Euler equations

I would like to write my own solver for compressible Euler equations, and most importantly I want it to work robustly in all situations. I would like it to be FE based (DG is ok). What are the ...
1answer
324 views

### How to construct well-balanced finite volume and discontinuous Galerkin methods for hyperbolic PDEs with source terms?

Source terms, such as those due to bathymetry in the shallow water equations, need to be integrated in a special way in order to preserve physical steady states. Is there a general way to construct ...
2answers
483 views

### Verification in Eigenvalue problems

Let us start with a problem of the form $$(\mathcal{L} + k^2) u=0$$ with a set of given boundary conditions (Dirichlet, Neumann, Robin, Periodic, Bloch-Periodic). This corresponds with finding the ...
1answer
905 views

### Can an approximated Jacobian with finite differences cause instability in the Newton method?

I have implemented a backward-Euler solver in python 3 (using numpy). For my own convenience and as an exercise, I also wrote a small function that computes a finite difference approximation of the ...
4answers
865 views

### Scalability of Fast Fourier Transform (FFT)

To use the Fast Fourier Transform (FFT) on uniformly sampled data, e.g. in connection with PDE solvers, it is well known that the FFT is an $\mathcal{O}(n\log(n)$) algorithm. How well do the FFT scale ...
1answer
398 views

### What spatial discretizations work for incompressible flow with anisotropic boundary meshes?

High Reynolds number flows produce very thin boundary layers. If wall resolution is used in Large Eddy Simulation, the aspect ratio may be on the order of $10^6$. Many methods become unstable in this ...
1answer
530 views

### Using fixed point iteration to decouple a system of pde's

Suppose I had a boundary value problem: $$\frac{d^2u}{dx^2} + \frac{dv}{dx}=f \text{ in } \Omega$$ $$\frac{du}{dx} +\frac{d^2v}{dx^2} =g \text{ in } \Omega$$ $$u=h \text{ in } \partial\Omega$$ My ...
2answers
7k views

### Periodic boundary condition for the heat equation in ]0,1[

Let us consider a smooth initial condition and the heat equation in one dimension : $$\partial_t u = \partial_{xx} u$$ in the open interval $]0,1[$, and let us assume that we want to solve it ...
1answer
3k views

### 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 ...
2answers
2k views

### Understanding the cost of adjoint method for pde-constrained optimization

I'm trying to understand how the adjoint-based optimization method works for a PDE constrained optimization. Particularly, I'm trying to understand why the adjoint method is more efficient for ...
1answer
3k views

### Strong vs. weak solutions of PDEs

The strong form of a PDE requires that the unknown solution belongs in $H^2$. But the weak form requires only that the unknown solution belongs in $H^1$. How do you reconcile this?