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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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6 views

Is this the correct way for solving coupled 1d PDEs using finite difference methods?

I am trying to solve the following coupled PDEs: $$C_e\frac{\partial u(x,t)}{\partial t} = k_{ed}\frac{\partial^2u(x,t)}{\partial^2x} - G_{el}(u(x,t) - v(x,t)) + S(x,t)$$ $$C_l\frac{\partial v(x,t)...
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0answers
47 views

MHD - How to impose a solid, perfect insulator as a boundary condition?

Consider the following MHD equations: $$\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\vec{u})=0,$$ $$\rho\frac{D \vec{u}}{Dt}=\vec{j}\times\vec{B}-\nabla p,$$ $$\frac{\partial\vec{B}}{\partial t}=\...
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2answers
106 views

How does a stiff equation solver work?

I am trying to understand how stiff differential equations are solved. For instance the equation, $$\frac{\partial y}{\partial t} = \alpha\frac{\partial ^2 y}{\partial z^2}$$ can be solved using ...
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0answers
63 views

Finite difference method for conservative form of equations

My question is about how do we discretize the equations in the conservative form using finite difference method. I'm trying to solve Euler equations in conservative form. $$ \frac{\partial u}{\...
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0answers
60 views

Understanding how to solve DAE

I am solving the following pde that is discretized in space using method of lines, in MATLAB using ode15s. $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\...
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1answer
89 views

Analytical Solution of Transport Equation

I'm looking at the analytical solution of the convection-diffusion equation $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$ with initial ...
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0answers
50 views

Implementing boundary condition

I'm studying the transport of species A in the blood vessels, $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$ At x=0, I want to use the ...
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0answers
38 views

Finite difference coupled PDE solution to 1D heat-reaction equation

I am trying to solve a coupled PDE for a thermal runaway reaction using finite difference method. I have 2 variables, temperature (T) and concentration (c) that vary as a function of time (t) and ...
1
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1answer
34 views

Radiation heat transfer between surfaces

I'm trying to model the temperature distribution over a curved surface. Apart from the heat equation, I need to take into account the energy emission/absorption through electromagnetic radiation. The ...
-1
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0answers
40 views

Choice of initial condition

I am trying to simulate the following system. $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$ with initial condition $$c(x,0) = C_o$$ and ...
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1answer
49 views

Numerical solution of non-linear first order partial differential equation (HJB)

I am trying to solve a simple optimal control problem using the Hamilton-Jacobi-Bellman equation, numerically in Python. This is proving to be rather difficult as I end up having to solve the ...
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1answer
46 views

Changing the domain of a 3D Finite Difference code from cube to sphere

I have an explicit FD (Finite Difference) code for diffusion/heat on a PDE in a cuboid domain, and it works fine. I would like to update the discretized equations and change the code so as to solve ...
2
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0answers
52 views

Identify the components of the (weak form) PDE in structural mechanics

I am trying to identify the weak form of PDE in structural mechanics. I read a lot of papers where they are using the elliptic boundary value problem \begin{equation} \int\limits_{\Omega} \delta \...
2
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0answers
61 views

2d wave equation with finite differences blowing up

I am (naively) trying to solve the 2d wave equation with finite differences. But the system blows up instantly. For simplicity I set the constant $c=1$, then I am left with $$\Delta u =u_{tt}.$$ I ...
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0answers
31 views

Guide for finite-difference schemes for Hamilton-Jacobi-Bellman Equations

I need to solve a simple, low-dimensional Hamilton-Jacobi-Bellman equation. Is there a simple guide for doing this numerically using finite-difference schemes? I found a few research articles ...
4
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1answer
96 views

Matrix Representation of a Discretization for a Partial Differential Equation

I want to discretize the following problem \begin{cases} \mu \nabla^2u+(\lambda+\mu)\nabla \nabla\cdot u = \rho \frac{\partial^2u }{\partial t^2 } + \beta \frac{\partial u}{\partial t}\\ u(...
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2answers
113 views

Method to find PDE equation coefficient satisfying mean solution?

What is the best approach to go about solving a PDE problem of the type \begin{equation} k^3\Delta u - k(\mathbf{1}\cdot\nabla u) = 0\, ,\\ u=g\; \text{on}\; \Gamma_D\, ,\\ mean(u) = u_\text{...
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1answer
132 views

Libraries to deal with unstructured grids

I am dealing with a *.cgns file. This mesh format, when saved as an unstructured grid, holds nodes coordinates, nodes connectivity per element and boundary ...
3
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2answers
89 views

How to simulate basic semiconductor models using the Drift-diffusion model on Python?

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 ...
1
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0answers
81 views

Finite differences for the one-phase Stefan problem

I am trying to code the one-phase, one-dimensional Stefan problem using finite differences in Matlab, similarly to what has already been done in Mathematica (see https://mathematica.stackexchange.com/...
2
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1answer
75 views

Stability Analysis

The partial differential equation, \begin{align} \dfrac{\partial f}{\partial x} + a(x)\dfrac{\partial f}{\partial y} = 0 \qquad & f(0,y) = f(L_1,y) = c_0e^{-y} \\ & f(x,0) = c_0 \;,\; f(x,L_2) ...
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0answers
64 views

Kernel independent fast multipole method for Yukawa potential [closed]

Has anybody used the KIFMM (https://web.stanford.edu/~lexing/fmm.pdf) for the Yukawa potential?
2
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0answers
27 views

Neumann boundary conditions in the Maccormack scheme

I am trying to solve the viscous Burger equation $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \mu \frac{\partial^2 u}{\partial x^2} $$ with Neumann boundary conditions. I am ...
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1answer
75 views

Multi-steps method for Navier-stokes equations with strongly nonlinear diffusion

I am trying to solve a particular form of the Euler / Navier-Stokes equations in 1D, with very strong and non-linear diffusion coefficients. My system of equations is \begin{cases} \...
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1answer
130 views

Does a generic method for solving a system of PDEs exist?

There are generic methods for solving systems of ODEs numerically. Are there generic methods for PDEs? If so, what are they? If not, why not? To elaborate... Any set of ODEs can be written in ...
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0answers
70 views

Solving a PDE implicitly by iteration in python

Connected to this question here on Computational Science, I've posted a follow-up question on how to solve a PDE using an implicit scheme like Crank-Nicholson in general in this question on SO. But I ...
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1answer
75 views

Simulation-based Optimization vs PDE-constrained Optimization

What is the difference between Simulation-based Optimization and PDE-constrained Optimization? Would studying a text on Simulation-based optimization be sufficient to understand and apply both?
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1answer
240 views

Solving PDE implicitly or explicitly depending on stiffness

I've got a system of several PDEs for a multitude of parts which represent real hydraulic parts like pipes or thermal energy storages. Each of these parts may have an arbitrary number of nodes and/or ...
5
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1answer
133 views

Finite Differencing schemes for Convection-Diffusion equation

I'm using the Convection(/advection)-Diffusion(-Reaction) equation to calculate the temperature over time in different hydraulic parts like a pipe or a heat exchanger. The flow/convection is always 1D,...
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0answers
79 views

Basic approach for numerical solution of PDE

I'm looking for some guidance on how to write a program to numerically solve a PDE. As an example for comparison in 1D: $$\frac{d^2u}{dx^2} = f\;\;\;\;u(0) = 0\;\;\;\;u(1) = 0$$ We could try 6 ...
2
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1answer
60 views

Is there Von Neumann stability analysis for 9-point laplacian like we have for the 5-point Laplacian?

For spatial accuracy in 2-D Laplace equation, a 9-point stencil is better than a 5-point one. $$\partial_tq= r\left(\partial^2_x q + \partial^2_y q\right)$$ for FTCS (forward-time, central-space) ...
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0answers
194 views

Need an explanation of setting Boundary Condition in pdepe (PDE Solver)

I am trying to solve the Advection-Diffusion equation using pdepe (PDE solver) in Matlab. I am not clear why we are writing ...
2
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1answer
50 views

Oscillation term in a posteriori error estimator

Assume that in the a (residual type) posteriori error estimator of some PDE is a term of the form $h_T\|g\|_{L^2(\Omega)}$ involved where $h_T$ is the diameter of an element and $g$ is some known data ...
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0answers
49 views

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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0answers
37 views

Simplest meaningful PDE/FEM calculation for mechanical stress due to heat

W have a complicated structure on which we do some FEM calculations regarding electrical potentials and heat distribution. The equations have the form $\nabla\kappa\nabla u = f + g\rvert_{N}$ where $...
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1answer
67 views

General questions regarding stability for time-integration of operator-split PDE systems

I am interested in solving ODE systems of the form \begin{align} \frac{\partial \vec{u}}{\partial t} = F(\vec{u}) \end{align} where $F$ is a nonlinear operator, $\vec{u}$ is a vector valued function ...
2
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1answer
106 views

Von Neumann's stability analysis on non linear and coupled equations

I'd like to know if is possible to make a Von Neumann's stability analysis on an system of coupled equations, featuring quadratics: $$\begin{aligned} \frac{\partial u_1}{\partial t}&=D_1\Delta ...
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1answer
69 views

How to simulate thermal expansion in a 2D plane using FEA?

I am trying to model 2D thermal expansion of a square area inside another square using FEATool. I have simulated plane strain by incorporating forces pointing along the $[1 \,\,\, -1]^T$ direction ...
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0answers
74 views

Finite difference methods for coupled 2nd order nonlinear pdes

I have a system of coupled nonlinear PDEs that I cannot figure out how to solve in a smart way using FDM, so I was hoping someone here might have a clue. The equations go as: \begin{align*} \frac{1}{...
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0answers
67 views

Discretize master equation

I'm trying to solve the following equation numerically using finite difference method. $\frac{\partial P}{\partial t} = -\frac{\partial}{\partial x_1}[F_1(x_1,x_2)P] - \frac{\partial}{\partial x_2}[ ...
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0answers
53 views

Why is my BTCS diffusion PDE solver matlab code not working?

I am trying to create a function that solves one BTCS step of the following PDE: $\frac{\partial q}{\partial t} = k\frac{\partial^{2}q}{\partial x^{2}}$ over a domain of: $0\leq x \leq \pi$ with ...
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2answers
126 views

Residual norm of PDE discretization: correspondence in the continuous problem?

Solving a linear PDE like $$ \Delta u = f \quad\text{on } \Omega,\\ n\cdot \nabla u = 0 \quad\text{on } \Gamma, $$ with Finite Elements usually goes like this: Create the discretization $Au=b$ via $$ ...
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1answer
60 views

Conjugate Gradient for non symmetric matrix

I have a large sparse matrix which is symmetric for the location of non zero values, but the values are different. Could I still use the CG method? I don't have much knowledge of linear algebra, the ...
3
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1answer
116 views

How to numerically minimize a functional?

How to numerically minimize a functional, for example, $$J[y]=\int_{x_1}^{x_2}L(x,y(x),y'(x))dx$$ An equivalent problem is to solve the Euler equation for this functional as a differential equation. ...
2
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1answer
82 views

Does the PDE hold at every cell in a FVM mesh?

If you solve a given PDE (Navier stoke's, Euler, heat eqn, advection eqn, etc...) using FVM, is this PDE supposed to be valid at every cell in the discretized domain, or only in the global domain as a ...
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0answers
47 views

Jacobian Elements for Coupled Drift-Diffusion System using Vertex-Centered Finite Volume

I'm trying to solve the fully coupled drift-diffusion system using Newton's Method. Although I eventually plan to potentially use a Jacobian-Free Newton-Krylov approach, this is still something that I ...
3
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2answers
394 views

Implementing no-flux boundary condition reaction-diffusion PDE

I'm having trouble figuring out how to implement boundary conditions for this problem: \begin{align} \frac{\partial n}{\partial t} &= D_n\nabla^2n - \nabla\cdot\left(\frac{\chi}{1+\alpha c}n\nabla ...
4
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2answers
170 views

treating “almost linear” nonlinear least-squares problems

As a model for a nonlinear least-squares problem with a large linear part problem, consider $$ \Delta u = 0 \quad\text{in } \Omega,\\ n\cdot\nabla u = 0 \quad \text{on } \Gamma,\\ (u(x_i) - u(y_i))^2 =...
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2answers
107 views

Numerically solving a PDE on an unit tangent bundle

Let $M$ be a manifold and $UT(M)$ its unit tangent bundle. I have a PDE which looks something like $$ \frac{\partial f}{\partial t}(x,v) = (v\cdot \nabla_x) f(x,v) + \Delta f(x,v) $$ where $x\in M$, $...
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0answers
79 views

Monotone, monotonicity preserving, LED, TVD, NVD, bounded, stable and stability preserving discretization schemes [closed]

When it comes to discretization schemes for finite volume method, the following terms can be found in literature: monotone schemes monotonicity preserving schemes local extremum diminishing schemes ...