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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17 views

Gauss Seidel moving mesh AMR hamilton Jacobi

I was trying to implement a moving mesh algorithm using Gauss Seidel, so: I have a pde like this (Hamilton - Jacobi eq) $\begin{cases}\phi_t+H(\phi_x)=0 & [-1,1]\times [0,T]\\\phi(x,0)=\phi_0 &...
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0answers
27 views

Relaxation Parameters for Steady Navier-Stokes

I am working on a project involving steady solutions for the Navier-Stokes Equations. In the past I've only worked with the unsteady Navier-Stokes, so some of this is new to me. In particular, at ...
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0answers
48 views

Can I model laminar incompressible fluid flow and heat transfer in MATLAB's PDE toolbox?

I have a system of PDEs in cylindrical coordinates that needs to be solved: 1. Continuity equation 2. Incompressible Navier stokes ( in r & z coordinates) 3. Heat transfer equation with both ...
9
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1answer
127 views

Adaptive gradient descent step size when you can't do a line search

I have an objective function $E$ dependent on a value $\phi(x, t = 1.0)$, where $\phi(x, t)$ is the solution to a PDE. I am optimizing $E$ by gradient descent on the initial condition of the PDE: $\...
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0answers
24 views

nonlinear coupled pde by finite difference

I want to solve nonlinear coupled PDE by finite difference method. I have done the discretization. the number of variables and number of equations are not same. How to deal with this situation?
2
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1answer
74 views

Coupling Boundary Condition of one PDE with source term of another PDE

We have a system of equations, wherein the BC of one PDE is coupled with the source term of another PDE. We have a regular 2D unit grid in x and y. There are two PDEs to be solved The first PDE (...
2
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0answers
40 views

Spectral/hp-finite elements for 4th order PDEs

Does anyone know of references discussing the solution of 4th order PDEs by way of spectral/hp-finite element methods? Specifically, I'm interested in the extension of the spectral element method, ...
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2answers
87 views

Help implementing finite difference scheme for heat equation

I am trying to solve the following problem via a finite difference approximation: $u_t = k \, u_{xx}$, on $0 < x < L$ and $t > 0$; $u(0,t) = u(L,t) = 0$; $u(x,0) = f(x)$. I take $u(x,0) = ...
3
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1answer
144 views

Numerical Methods for Solving a Fully Nonlinear Time-Dependent PDE?

Are there numerical methods of solving the following fully nonlinear time-dependent PDE: $$\nabla^2u\left(\textbf{r}(t), \dot{r}(t), t\right)=f\left(\textbf{r}(t), \dot{r}(t), t\right),$$ for $\textbf{...
3
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2answers
72 views

Eigenvectors of Black-box matrix

$\DeclareMathOperator{\diag}{diag}$ Consider the generalized eigenproblem $A\mathbf{x}=\lambda B\mathbf{x}$. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet ...
2
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0answers
11 views

Dynamic Successive Over/under Relaxation (SOR) with several variables

I am solving a partial differential algebraic equation (PDAE) system which has the following dependent variables: $f=f(X,T)$ and $g=g(T)$, along with a few others My current method for coupling is ...
4
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2answers
75 views

Discretizing time term for PDEs

I am actually confused about time stepping for PDEs. Before, I was distretizing time using backward Euler method for implicit formulation and I get a system of Algebraic equations to solve. Now during ...
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0answers
51 views

Reaction-diffusion equations

I'm simulating a biological phenomena with reaction diffusion equations. There are multiple diffusing materials and there are some complex relations about consumption and production of such materials. ...
4
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0answers
98 views

Effects of Lumping Mass Matrix

I've recently finished an introductory course on the finite element method from a more mathematical perspective (following Brenner and Scott) and we were introduced to the finite element mass matrix ...
0
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0answers
31 views

Stiff ODEs coupled with PDEs (computational efficiency)

I am simulating in COMSOL a system of 3 coupled PDEs (parabolic & elliptic) along with 10 stiff ODEs. In order to have the system working, I am downsizing the time step size too much to achieve ...
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0answers
26 views

Solving nonlinear coupled PDE using FiPy

I have recently been trying to solve these 6 coupled, nonlinear PDEs of the general form: $ \frac {\partial N_1}{\partial t} = -a(P_2E_1 + P_1E_2) - bN_1 \\ \frac {\partial N_2}{\partial t} = -c ...
3
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2answers
77 views

Numerical solution of burgers equation with finite volume method and crank-nicolson

I'm having difficulty with numerically solving the inviscid burgers equation.Godunov's scheme is used in most of what I've found in literature . Now my question is if using a crank nicolson shceme is ...
3
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1answer
97 views

Applicable solvers for nonlinear coupled PDEs

I've been trying to find an applicable PDE solver for cases such as this: Although when dealing with stiff equations in the complex domain, applying existing packages has been problematic. I've ...
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0answers
31 views

PDE solver for drift-diffusion with generation

I am trying to solve the drift-diffusion model with generation, $$\frac{\partial N_e}{\partial t} = \alpha(x) \left| \Gamma_e (x,t) \right| - \frac{\partial \Gamma_e}{\partial x} (x,t)$$ $$\frac{\...
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0answers
35 views

Finite differences for incompressible viscous fluid equations

I am working with the equations for incompressible viscous fluid: $$ \partial_t \vec{\omega} + (\vec{u}\cdot\nabla)\vec{\omega} = \nu\nabla^2\vec{\omega} $$ $$ \nabla^2 \vec{\psi} = -\vec{\omega} $$ $...
3
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1answer
128 views

Achieving high relative accuracy (vs. absolute accuracy) using spectral methods

Problem I have a PDE that I'm trying to solve with spectral methods. The solution $y$ is always positive, and decays as $y \propto e^{-ax}$ for large $x$. The domain is $[0, \infty)$. (There are ...
5
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2answers
108 views

Is this system of diffusion equations well-posed?

I’m using a standard Crank-Nicholson algorithm to solve this system of two coupled diffusion equations: $\dot{u} - \dot{v} = \frac{\partial}{\partial x} \left( \alpha(x) \frac{\partial u}{\partial x} ...
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0answers
40 views

Manufacturing a solution for non-smooth coefficients in elliptic problems

This question is a continuation of this answer (I can't comment) If we were going to manufacture a solution for a problem with discontinuous coefficients, I understand that the solution should have ...
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0answers
31 views

Solving Laplace's Equation in Cylindrical Coordinates in Mathematica

This is my first attempt at solving a PDE with boundary conditions numerically, and I'm not sure if Mathematica has that functionality built-in. Basically, I am trying to solve Laplace's equation in ...
2
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1answer
92 views

conservative v non-conservative [duplicate]

I have always come across the terms "conservative" as opposed to "non-conservative" forms of equations in fluid mechanics. Is there a good reference that someone can share to clearly distinguish the ...
0
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1answer
88 views

How do I solve Laplace's equation in 2D using spectral methods?

I want to solve the 2D Laplace's equation: $$ \frac{\partial^2 T}{\partial x^2 } + \frac{\partial^2 T}{\partial y^2 } = 0 $$ with boundary conditions: T(x=0)=T(x=1)=T(y=1)=0 and T(y=0)=1 on a ...
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0answers
18 views

Second and Higher Order Order Corrector in Spectral Deferred Correction

I am trying to work out a second order or higher order correction for the method of Spectral deferred Correction (SDC). Specifically using as a corrector a second order or third order multi-step. In ...
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0answers
23 views

How come the use of delay differential equations in model parameter estimation better than ordinary differential equations? [closed]

in systems biology why is the use of delay differential equations better than ordinary differential equations i.e. compartment models in delay modelling? is there a data independence angle in models ...
0
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1answer
56 views

Solving coupled differential equations with multiple independent variables

I am currently using the wrapper odeintw for scipy.integrate.odeint to solve my equations since they are complex-valued. At the moment, I have 3 coupled first-order differential equations with 2 ...
5
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0answers
84 views

Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \begin{equation} \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - \...
1
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1answer
45 views

FreeFem user-defined function [closed]

I'm trying to solve the steady-state heat equation with a space-dependent thermal diffusivity in FreeFem. I.e. $$\nabla\cdot(\kappa(x,y)\nabla T) = 0$$ I'm hoping to read in from a file locations $(...
5
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0answers
36 views

Phase dislocations and numerical accuracy

I am solving the nonlinear Schrodinger equation (NLSE), $$A_t+iA_{xx}+i|A|^2A=0$$ where $A$ is a complex valued function, which can be written as $A=ae^{i\theta}$ for $a,\theta$ real. Now, for ...
4
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1answer
122 views

High order unconditionally stable discretization for a scalar hyperbolic PDE

In order to numerically solve the following differential equation: \begin{equation} \text{Fr}\{f\} := v(k)\dfrac{\partial f(z,k)}{\partial z} - F(z) \dfrac{\partial f(z,k)}{\partial k} = -\dfrac{f-...
1
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2answers
62 views

The second variation of displacement interpolation function in Finite Element Method

I need to calculate the second variation of displacement interpolation function $u = \sum N_a u_a$ in Finite Element Analysis, where $N_a$ are the shape functions and $u_a$ are the nodal values. ...
4
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1answer
67 views

What is the preferred method for evolving the Nonlinear Schrödinger Equation?

I am interested in evolving the (cubic) self-focusing nonlinear Schrödinger equation, $$i\frac{\partial \psi}{\partial t}+\frac{1}{2}\frac{\partial^2 \psi}{\partial x^2}+\left|\psi\right|^2\psi=0$$ ...
3
votes
1answer
125 views

Is this finite difference approach correct?

I am solving incompressible 2D Navier-Stokes equations with zero y-component velocity. The geometry is a simple 2D pipe of a length $L$ and diameter $W$ and there is only two boundary conditions: ...
0
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0answers
85 views

How to solve a set of 1st order 1D PDEs in Matlab?

i have already asked something similar to this once but sadly i wasn't able to get a complete answer, so i remain stuck on this problem. the Matlab function pdepe seems to only be able to solve 2nd ...
0
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1answer
46 views

How to use time delays in the solvepde function in MATLAB for a system of PDEs?

The solvepde function was introduced in MATLAB R2016a. I am able to solve my system of PDEs if there are no time delays involved. Does anyone know how to include time delays in the solvepde function?
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22 views

Convergence of the stress at an interface using AMR with quadtree meshes in a solid mechanics problem

Assuming a solid mechanics problem, linear elasticity, with a domain split in two by an interface that is not aligned with the mesh. The mesh is a quadtree with squares. There are different material ...
0
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1answer
166 views

How to solve my PDE's in Matlab?

I need to solve a set of 5 PDEs for functions $u(x,t)$. I looked up the Matlab function pdepe. It looked perfect for my case, until I read the line: $f(x,t,u,\...
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0answers
60 views

Stability of two PDEs

I have the following two PDEs which I want to check for stability: $$u_t= u_{xx} , \ u(x,0)=1 , \ u_x(1,t)=-hu^4(1,t) , \ u_x(0,t) = 0 $$ $$u_t = u_{xx}-\sin(x+t)+\cos(x+t) , \ u(x,0)=\cos(x) ,\ u_x(...
2
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1answer
123 views

How do I program periodic boundary conditions? [duplicate]

Hi I have a code below that solves non linear coupled PDE's given Dirichlet boundary conditions. However I need to implement periodic boundary conditions. The periodic boundary conditions are ...
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1answer
89 views

Stability analysis for coupled nonlinear system of partial differential equations

I'm trying to solve a nonlinear partial differential equation \begin{equation} L(u_{xxtt},u_{xx}u_{tt},u_{xt}^2,u_{xt},u_{tt})=0 \end{equation} using finite difference methods. In order to remove the ...
6
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1answer
174 views

What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I am reading a paper [1] where they solve the following non-linear equation \begin{equation} u_t + u_x + uu_x - u_{xxt} = 0 \end{equation} using finite difference methods. They also analyse the ...
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3answers
205 views

Algorithm suggestion for PDE - example: heat equation

I want to solve the PDE equation numerically. For this, I started my study with something simple; heat equation $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial^2 x} $$ with the initial ...
4
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1answer
58 views

Boltzmann equation and equilibrium distribution function

The free-streaming term in stationary 1D Boltzmann equation satisfies the equilibrium distribution function, that is: \begin{equation} \text{Fr}\{f\} := v(k)\dfrac{\partial f(x,k)}{\partial x} - \...
3
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0answers
39 views

Convergence of KKT equations for discrete parameter estimation problems

Consider a discrete constrained optimization problem: $$ \mathbf{q}_*^h= \arg \min {\cal J}^h(\mathbf{x}^h[\mathbf{q}^h],\mathbf{q}^h) $$ subject to the (weak-form) constraint $$ F^h[\mathbf{x}^h(\...
0
votes
3answers
319 views

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
91 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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0answers
60 views

Solving system of equations with zeros on diagonal [closed]

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} + A$...