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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Numerical simulation of a reaction-diffusion system on MATLAB with finite difference discretization of spatial derivative

I have a model of a system which consists of diffusing reactants and intermediates. For each variable, $u_i$, the final representative equation looks like the standard reaction-diffusion form: ...
3
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0answers
80 views

Symplectic integration of PDE

I consider ordinary wave equation $$ u_{tt} - u_{xx} = 0 $$ with initial conditions $$u(x, 0) = \exp (-2 x^2) \\ u_t(x, 0) = 0 $$ To solve this problem I approximate $u_{xx}$ with 4-th order ...
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49 views

Solving system of equations with zeros on diagonal [on hold]

I am using finite elements, Newton Raphson technique and MUMPS direct solver, to solve for $P$ in this equation: $$\frac{\delta K}{\delta x} \frac{\delta P}{\delta x}= \frac{\delta K B}{\delta x} + ...
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40 views

PDE discretization (via finite difference sheme) question

So after posting this question and reading all your comments I would like to make this new question (update). If you consider the three equations presented here: $$\frac{\partial \rho}{\partial t} ...
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1answer
33 views

Non-uniform finite difference Adaptive Mesh Refinement

Assuming that the crosses in the figure below are unknowns in a vertex-centered finite difference scheme in an Adaptive Mesh, how can I calculate the double derivate (Laplacian) at the Red x ? The Red ...
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1answer
49 views

Solving coupled PDE in COMSOL [closed]

I have the system of equations \begin{align} &A \frac{\partial u_1}{\partial t} = 1 - u_1 B \frac{\partial u_2}{\partial y}\\ &\frac{\partial u_2}{\partial t} = \frac{\partial}{\partial ...
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0answers
22 views

Entering an Electric field dependent equation in material contents in COMSOL?

I am trying to model a CNTFET in COMSOL. I am following the same steps that were used to model a MOSFET (provided in the semiconductor module manual), except for a few additional layers and materials. ...
1
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1answer
70 views

How to deal with PDE over the real line

I have a PDE defined over $\mathbb{R}$, for which I don't have the exact solution, and I am to approximate it with finite differences so I need to input some BC. Can anyone suggest any good ...
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0answers
36 views

Error norm in maple

I have the function $cos(x+t)$; the following PDEs have $cos(x+t)$ as the analytical solution of them: $$u_t = u_{xx} -\sin(x+t) + \cos(x+t) \\ u_x(0,t)=-\sin(t) \\ ...
6
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1answer
135 views

A simple PDE solution question

I need to ask a question about partial derivatives. I want to solve this equation (steady state, one dimensional continuity equation): $$\frac{\partial (\rho u)}{\partial z}=0$$ which is equivalent ...
4
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1answer
69 views

Refinement in AMR

Assume I start with an 8x8 coarse mesh (see Fig 1) where the vertices (except boundary vertices) represent the unknown variable. After iterative approximation - ...
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2answers
56 views

Fourier techniques and periodic boundary conditions

Could somebody explain to me why periodic boundary conditions are automatically satisfied if you solve your problem assuming a Fourier series? So, if we assume a Fourier series for our solution, we ...
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0answers
48 views

Simulation of parameter propagation of a numerical PDE solution

I have the following PDE: $$u_{xx}u^3 - \sin(xt)u_{tt}=u(x,t)$$ $$ u(x,0) = h,\ u_t(x,0) = 1-h,\ u_x(0,t)= h*u^3(0,t),\ u_x(1,t)= 1$$ where the problem is defined for $t>0$ and $x \in [0,1]$, and ...
7
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1answer
103 views

Solving a simple Schroedinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Piaevskii equation I have found a problem that can be tracked down to something buggy occuring in the simplest Schrodinger equation possible: $\partial_t \psi ...
3
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1answer
73 views

Comments needed on the doubts of PDEs in moving boundary problems

We know that in classical two-phase Stefan problems, let's say in the temperature distribution of ice-water problem here, the governign PDEs are: \begin{equation} \left. \begin{aligned} ...
2
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2answers
113 views

Are the Finite Difference and Finite Volume Methods different after the application of the Gauss Divergence theorem on the FVM?

I have a rudimentary question about the differences between finite difference (FDM) and finite volume methods (FVM). In FDM we concentrate on the nodes (points) in space while in FVM we concentrate ...
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0answers
85 views

Sobolev space H^2

In the definition of $H^2$ what does mean that the first and the second partial derivatives of some $u$ belonging to this space belong to $L^2$ in the meaning of distributions?
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1answer
68 views

Convection diffusion reaction equation (stiffness, solver)

I am trying to solve the CDR-Equation in 2D: $$\frac{\partial c(x,y)}{\partial t} + \nabla \cdot ( -d\nabla c(x,y) + \vec{v}(x,y) c(x,y))+ a c(x,y)=0\,,$$ with Boundary Conditions (length of square ...
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1answer
43 views

Looking for a matlab/maple code for plotting the truncation error

On page 18 on this text: http://www.dima.uniroma1.it/users/lsa_adn/MATERIALE/FDheat.pdf , the graph in figure 8 on this page, how would I write a suitable code in matlab or maple that will produce ...
8
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1answer
184 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?
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1answer
48 views

Implementation of Neumann boundary condition with method of lines - 1D diffusion/reaction equation

I am solving the monodimensional diffusion/reaction equation by discretization using the well-known method of lines ${\partial c\over\partial t}=D{\partial^2 c\over\partial x^2}+ r\text{ for ...
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19 views

Adsorption in chemical engineering

Is there special method to solve adsorption equations in chemical engineering? I mean a mixture of gas pass through a adsorbent and special gas adsorbed.
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1answer
125 views

How can I solve stiff equations by method of line (MOL)?

I want to solve 7 coupled equations.I use method of line(MOL) and discrete the equation in Length and radius and convert them to a system of ODEs in time.and use ode15s to solve them in MATLAB. But an ...
0
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0answers
44 views

Solving a nonlinear problem with CDF

I'm trying to solve this problem: $\begin{cases} \partial_t E=-k\left([f(\rho)-i.\left[\delta+\frac{1}{2}a\left[\dfrac{\nabla^2_{\bot}}{4}+1-\rho^2\right]\right]]E - 2CP\right)\\ \partial_t ...
5
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1answer
106 views

PDE - Conservative form, conservative methods and discrete conservation

I cannot find a reference explaining clearly and rigorously the links between the notions of conservative form for a PDE, a conservative numerical method and discrete conservation. I would be very ...
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2answers
115 views

Solving coupled PDEs numerically on a semi-infinite domain with no-flux boundary conditions

I have the following system of PDEs for which I have given parameters $\gamma, \tau$ and $\mu$, $$\begin{align} T_t = &\ \gamma\,(L +\tau F-T)\\ F_t = & -F_x-(F-LT)\\ L_t = &\ \mu ...
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0answers
80 views

System of PDEs in Julia [closed]

I'm having difficulty coding and was hoping someone could see my error. I'm using Julia with the ApproxFun package and am trying to solve the wave equation with Dirichlet boundary conditions ...
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0answers
54 views

Solving Laplace equation on complex geometry using mixed boundary conditions

Consider a regular, Cartesian grid. The domain where I want to solve Laplace equation consists of a number of grid points on the grid. They form a complex geometry. I want to solve the Laplace ...
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0answers
18 views

Duality/Lagrangian condition and Variational Inequality of a cost functional

Given the functional $J_A(y,u) = \frac{1}{2}||y-y_d||^2_{L^2(w,\omega)} + \frac{\lambda}{2}||u||^2_{L^2(w^{-1}, \omega)}$ where w is a function belongs to Muckenhoupt class. Given the optimization ...
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0answers
45 views

Nitsche' method for handling Dirichlet boundary conditions [duplicate]

I want some explanations about the genesis of Nitsche' method Thanks
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0answers
78 views

Boundary Conditions for the given PDE

I'm working on the Black-Scholes equation, but I'm pretty new to financial modeling. Right now, I am trying to understand the Black-Scholes PDE. I understand that the Black-Scholes equation is given ...
0
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1answer
107 views

Heat Equation - PDE

I'm trying to model the Black-Scholes Equation (transformed into a heat equation) using method of lines in Python. The transformed formula is basically \begin{equation*} \frac{\partial u}{\partial ...
2
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0answers
69 views

Modeling First Order Parabolic PDE (Battery Storage Model)

I'm trying to solve the following first order parabolic partial differential equation, \begin{equation*} X \frac{\partial V}{ \partial Q} = -\frac{1}{2} \sigma^2 \frac{\partial^2 V}{\partial X^2} + ...
2
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0answers
154 views

Unrealistic solution for advection-diffusion-reaction PDE with heterogeneous media

About the code: I have a code which simulates concentration from advection-diffusion-reaction PDE in 2D space (X,Y) with time. The solution is obtained using fully implicit finite-difference method ...
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0answers
81 views

Time discretization of wave equation

I am trying to model the seismic wave equation and have therefore been reading about discretization schemes and their stability. I recently came across an insightful paper on 'Galerkin FEM methods for ...
0
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2answers
116 views

Wave Equation PDE [closed]

I'm trying to solve the following PDE wave equation using method of lines: Wave Equation: u_tt = u_xx with initial condition: u(0,x) = sin*pi,u_t(0,x)=0, 0 < x < 1 boundary ...
3
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2answers
178 views

Hyperbolic Equation PDE (Python)

I'm trying to solve the following first order hyperbolic PDE problem using method of lines: Hyperbolic Equation: $u_t = -u_x$ with initial condition: $u(0,x) = 0, 0 < x < 1$ ...
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0answers
89 views

von Neumann stability analysis system of equations

How can I apply von Neumann stability analysis for the following system of equations? Discritization method: Lax method \begin{align} &\frac{\partial u}{\partial t} + a \frac{\partial v}{\partial ...
1
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1answer
140 views

Solving DAE with higher index

I want to solve 7 coupled differential equations in MATLAB. I used the method of lines (MOL) to convert them to a system of DAEs. I then attempted to solve this system using ode15s. Unfortunately an ...
2
votes
1answer
46 views

How can I analyze the stability of a PDE discretization at a boundary?

I have a numerical discretization of a partial differential equation that seems to be unstable or stable at a boundary point, depending on what finite difference scheme I am using. Are there standard ...
2
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1answer
68 views

Laplace's equation with periodic Dirichlet boundary conditions

Consider a Laplace's equation with Dirichlet boundary condition: ${\nabla ^2}\Phi = 0$ in a domain $D$ with given Dirichlet Boundary condition: $\Phi=\Phi_o$ at $\partial D$ (smooth, but not ...
2
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2answers
74 views

PDEs appropriate for adaptive time stepping algorithms

I'm looking for some physical phenomena for which an adaptive time stepping algorithm would be ideal. A PDE or ODE that showed very large gradients in time at a small period of time and smoother ...
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2answers
73 views

2d Euler manufactured solutions

Where can I find manufactured solutions for the 2d Euler equations, with the complete analytical terms, including the Jacobian of the source term ?
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2answers
63 views

Heat Equation Solution in One Dimension (x, t)

We're currently solving the heat equation as a part of the PDE sequence in class. We've been given the formula:$$T(i, n+1) = T(i,n)+\alpha \left [\frac{T(i+1,n)-2 T(i, n)+T(i-1,n)}{\Delta x^2} \right ...
2
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1answer
64 views

FDM - Solving acoustical wave equation via first order PDE's

Solving acoustical wave equation: $$ c_0^2\partial_{xx}p-\partial_{tt}=0 $$ using forward-time centered-space FDM is not very convenient cause of numerical dispersion etc. What about using a little ...
1
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1answer
124 views

How can I solve coupled equations by the method of line(MOL)?

I want to solve 3 coupled PDEs equations. They depend on time, radius and length. I used the method of lines (MOL) and converted them to a system of ODEs in time. Now I want to solve them using ...
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2answers
82 views

Mathematical test method for the numerical solution of PDEs?

What are some of the methods used to test for the exactitude of a numerical solution, given that the analytical solution isn't available, and the numerical solution converges ?
2
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1answer
59 views

Finite strain FEM using an existing code that solves small strain elasticity

I have an existing FEM code that solves the linear elasticity problem. I would like to use the same code for large strain rates, still using a simple material law (Saint Venant–Kirchhoff model). [The ...
3
votes
2answers
98 views

How to implement boundary conditions in heat equation with no flux and fixed value at the same time? Is it Robyn BC?

I am modeling the temperature of the groundwater using heat equation. I have Dirichlet BC at the top but at the bottom I have constant temperature equal 12 degrees C (see attached pic). It is look ...
5
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1answer
169 views

Do the class of PDEs that lack initial conditions have a name?

I am trying to think of what this kind of problem is called. I illustrate it with a telegrapher's equation with (hopefully) standard notation. Find $u:\Omega\times \mathbb{R} \to \mathbb{R}$ such ...