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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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1answer
67 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
62 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?
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0answers
19 views

Globalization strategy for semismooth Newton methods

Currently, I'm trying to implement a semismooth Newton method for a nonlinear problem of the form \begin{align*} a(y,v) + \langle F(y), v\rangle = \langle f,v\rangle \quad \forall v\in V \end{align*} ...
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0answers
19 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 ...
1
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1answer
68 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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0answers
32 views

On and off boundary condition

Hi I want to solve this problem using ode15s solver. Only issue I am not getting that is; this problem has a on and off boundary condition for specific time each day. To make it more specific say ...
1
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1answer
125 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 ...
4
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0answers
59 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 ...
0
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1answer
67 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?
5
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1answer
232 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
110 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
78 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
52 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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66 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
43 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
36 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 $...
1
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1answer
66 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
62 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 ...
0
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1answer
67 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
62 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
64 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
44 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 ...
5
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2answers
123 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
53 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
112 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
80 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
33 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 ...
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2answers
306 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
165 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 =...
5
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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
70 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 ...
1
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1answer
62 views

Physical data for heat equation

I want to implement an algorithm to solve a heat equation, i.e. \begin{align*} \partial_t u - \Delta u = f \text{ in } \Omega\times(0,T)\\ \partial_nu = 0 \text{ in } \partial\Omega \times (0,T)\\ u(0)...
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1answer
45 views

Trying to solve a wave-like equation

I'm trying to solve an equation whose solutions I know are plane waves but there are a few nuances. First, the equation is of the form $$ \partial^2_t \psi + A(r)\partial^2_r \psi +B(r) \partial_r \...
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0answers
93 views

How to account for the interface between two different phases in a discretized diffusion model?

I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a ...
0
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2answers
99 views

Second derivative in coordinate invariant form

To solve stationary, incompressible, inviscid and irrotational flow around a circular cylinder, I am using general coordinates. Since the flow is symmetrical, we only consider the upper half of the ...
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2answers
116 views

Mass conservation in 1d diffusion by method of lines

I am solving the 1D diffusion equation by discretization using the method of lines. My problem is that I don't manage to ensure mass conservation. I have read many similar questions about the topic ...
0
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1answer
192 views

Crank–Nicolson method for nonlinear differential equation

I want to solve the following differential equation from a paper with the boundary condition: The paper used the Crank–Nicolson method for solving it. I think I understand the method after googling ...
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2answers
116 views

Find classical solution of transport equation with FDM

We know the classical solution of transport equation is determined by one initial (boundary?) condition, for example, the solution of $$\frac{\partial u(t,x)}{\partial t}+\frac{\partial u(t,x)}{\...
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2answers
109 views

Nédélec Elements and Newton-Methods

If you want to develop numerical algorithms for variational inequalities, you often choose a Semismooth Newton Method. In many cases, this approach involves derivatives of $\max$ or $\min$ functions ...
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0answers
141 views

Methods and tools to solve the two-temperature model (TTM)

I would like to model heat diffusion at the gold / water interface after excitation of the metal surface by an ultrafast laser pulse (ca. 80 fs). An appropriate model to start with would be the "two ...
6
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1answer
153 views

Quantifying the degree of nonlinearity in a heat transfer problem

I’m working with a heat equation of the form. $$\frac{d(\rho(T)c_p(T)T)}{dt}-\nabla\cdot(k(T)\nabla T)=f$$ with temperature dependent density $\rho(T)$, specific heat $c_p(T)$, and thermal ...
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0answers
145 views

Analysis and numerical methods for PDEs arising in industrial problems

Assume some basic knowledge of numerical methods for PDEs, acquired through A. Quarteroni's Numerical models for differential problems. I'm looking for a reference to get started on the analysis of ...
3
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1answer
71 views

FEM problem: how to get a feeling for size of problem

The following problem is given: $$ - \bigtriangleup u = f, \quad \partial_n u = g \quad \text{on } \Gamma_N, \quad u = 0 \quad \text{on } \Gamma_D $$ with $\Gamma_N$ or $\Gamma_D$ denoting the ...
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1answer
93 views

Is the 1D wave equation analytically solvable for a Neumann BC? [closed]

The general solution to the 2nd order wave equation: $$\frac{d^2u}{dt^2} = c^2\frac{d^2u}{dx^2}$$ is known as d'alembert's formula->https://en.wikipedia.org/wiki/D%27Alembert%27s_formula Does an ...
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1answer
260 views

Proper boundary conditions for potential flow around cylinder

I am computing the stationary, incompressible, inviscid and irrotational flow around a circular cylinder using a discretization in general coordinates. I derived a PDE and proper boundary conditions ...
6
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3answers
185 views

Why naively chopped finite difference matrix works for different ODE boundary conditions

We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. One naive way to write down the matrix of the differential operator is like the following, ...
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0answers
58 views

Dealing with non-physical negative ODE solutions using ODEPACK

Hi and thank you all again. I am solving the reaction-diffusion-advection equation as follows $ \partial_{t} n\left(t,z\right) = -\partial_{z}\left(\Phi\left(t,z\right)\right) + p\left(t,z\right) -n\...
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0answers
103 views

How to apply an integrated constrain condition in FEM?

I'm running some simulation using FEM. In my model I need to apply a constraint condition to the governing equation. My governing equation similar to the diffusion equation as below: $$\frac{\...
3
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2answers
230 views

Developping PDE with Python symbolically and numericaly

I feel like publishing some previous works from my PhD thesis. I was using Mathematica to build a system of 2N partial differential equations for 2N functions by symbolic spatial Taylor expansion, ...