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
68 views

1D wave equation using Finite difference method MATLAB

I have the wave equation $$u_{tt} = 4 u_{xx}$$ with the boundary conditions $$u(0,t) = u(L,t) = 0\,,\quad x \leq 0 \leq 2\pi \,,\quad t\geq 0$$ and initial conditions $$\begin{align} &u(x,0)=\...
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0answers
37 views

Appropiate Artificial Boundary Conditions for the radial part of the Klein Gordon equation?

I am trying to simulate the following equation using FDTD $ \left(- \partial^2_t + \partial^2_x + V(x) \right) \psi(x,t) =0 $ subjected to the initial conditions $\psi(x,0) = f(x),~ \partial_t \psi(x,...
2
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1answer
377 views

Robin Boundary Condition with Implicit Upwind - Finite Difference Method for 2D Convection-Diffusion Equation

I am trying to solve a problem with 2D Convection-Diffusion equation with U = Concentration ($mg/m^{2}$) using Implicit Upwind Finite Difference Method like this $$ \frac{\partial U}{\partial t} + ...
6
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0answers
106 views

FEM : energy minimization VS PDE solving

Engineering FEM When I studied engineering, I learned the traditional approach for finite elements for elasticity. The point was to solve the PDE $-div(\sigma)=f$ as: Multiply your PDE with a test ...
3
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1answer
109 views

Wave equation, wave is bouncing off Neuman boundary

Wave equation. Mixed BC. Applied Neuman boundary condition ($\frac{\partial u}{\partial x}\big|_N=0$) to the RHS of the domain You may observe the sharp edge in the middle of the string in the image ...
2
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1answer
63 views

Finite element modelling of thermal expansion of 3D solid bodies

I want to solve the thermal expansion of solid by using FEM approach. When I developed the model based on the the principle the minimum potential energy, the solutions for thermal expansion are not ...
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1answer
537 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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1answer
141 views

Non-reflective boundary condition

I'm currently solving incompressible Navier-Stokes system of equations with periodic flow and high viscosity. Is there any outlet boundary types that avoids the reflection of flow from the outlet back ...
2
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2answers
297 views

Semi-infinite domain transformation

Question is mostly related to literature or suggestions. Given a semi infinite domain: $x=[0; +\infty);y=[0; +\infty)$. Willing to transform it to computational domain of: $[0,1]\times[0,1]$. I did ...
3
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1answer
451 views

Poisson image blending artifacts

I am trying to implement Poisson image blending as in the paper Poisson Image Editing. This is the task of filling in a masked region of an image by minimizing $$\min_f\int_\Omega \left | \nabla f - \...
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2answers
89 views

What is known about C0 triangular finite elements with nonstandard mesh point placement?

I'm curious about the general case, but for ease of explaining lets just take the case of a $P^2(\Omega)$ approximation. For simplicity, let's also just consider the reference element $(0,0), (0,1), (...
3
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1answer
78 views

"Optimal" domain partitioning in domain decomposition algorithms

When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
2
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1answer
184 views

Solving stiff ODEs: Dealing with Jacobian terms which take too long to compute with finite differences

I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
0
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1answer
340 views

Simple particle-in-cell examples

I am studying about the 1D EM-PIC (Electro Magnetics using particle-in-cell) simulation. I want to have a simultaneous time-integration of the electric/magnetic fields plus the motion of free charges ...
2
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1answer
115 views

Developing a meshfree contouring algorithm

I would like to find the contours of a scalar function $u(x,y)$ available as a discrete set of function values $u_i = u(x_i,y_i)$ over a scattered set of points $(x_i,y_i), i=1,...,N$. In my case, the ...
8
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1answer
4k views

Computational methods for finding the energy eigenvalues of the time-independent Schrodinger equation with arbitrary potential

I have seen in some papers that the energy levels in some arbitrary potential are specified. How can one find the energy levels in such arbitrary potentials. For example, $V(x)=\sin^2(x/2)$ with $x\in[...
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1answer
267 views

Numerical solution of non-linear heat-diffusion PDE using the Crank-Nicolson Method

I am trying to solve numerically the following 1D EBM: $C\frac{\partial T[x,t] }{\partial t} - \frac{\partial }{\partial x}\left ( D(1-x^2)\frac{\partial T[x,t] }{\partial x} \right ) + I[T] = S[x,t](...
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1answer
90 views

Stability analysis simplification for PDE

I have the nonlinear PDE $$\frac{\partial U(z,t)}{\partial t} + A(U)\frac{\partial U(z,t)}{\partial z} + B(U)U(z,t) + C(z,t) = 0,$$ where $A(U)$ and $B(U)$ are guaranteed to be real and positive. I ...
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0answers
49 views

What is the most common loss function used with collocation methods for differential equations

I was looking at the Cheney and Kincaid book (6th edition) on numerical methods, with respect to collocation method for differential equations. Now for linear systems of ODES, collocation is just a ...
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1answer
109 views

interface value on the error equation

https://www.jstor.org/stable/pdf/2157482.pdf, here I have a problem in last equation of (2.6) in section (2.1). When they are considering error equation on the interface $\Gamma$ they get $e_v^{(n)} = ...
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0answers
53 views

Discretizing Multi-species Ion Exchange Equations by Finite Volume Method

I'm solving a system of multispecies ion exchange equations (diffusion+drift fluxes) in 1-d spherical domain using finite volume method to obtain the ion concentrations at the next time step. After ...
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1answer
26 views

How to decrease error in (FTCS) forward time centered space method?

I am using the FTCS method for solving differential equations. I know that the condition for stable output is $$ \frac{\alpha \Delta t}{\Delta x ^2} < \frac{1}{2} $$ But when I use the distance ...
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0answers
50 views

How to numerically solve PDE that governs the free vortex wake model?

Crossposted at Math SE I am reading a paper on the free vortex wake model for a helicopter rotor blade, which is described by the following PDE: $$\frac{\partial \vec{r}}{\partial \psi} (\psi, \zeta) ...
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0answers
44 views

introduction to non-overlapping domain decomposition in 1D

I am new to domain decomposition and have been searching in google for an introduction in 1D which goes over the complete procedure from the continuous to the discrete problem as well as the algorithm ...
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2answers
181 views

Continuation of solution to $-\nabla\cdot (k(x,y)\nabla u)=f$

I'm trying to solve the following problem, I had previously opened another discussion for the implementation and well, it seems that it has turned out well, it can be found here. I need to calculate ...
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2answers
356 views

Implementing routine for $-\nabla\cdot (k(x,y) \nabla u)=f$ in Matlab

I am solving the Poisson Equation for 2D given by the following expression: $$-\nabla\cdot (k(x,y) \nabla u)=f$$ in a rectangle with Dirichlet conditions on the boundary using Matlab. In principle I ...
3
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0answers
80 views

Strange Picard iteration

I am interested in solving the equation $$ \begin{aligned} \nabla \cdot\left(\nabla \phi-\frac{\nabla \phi}{|\nabla \phi|}\right) &=0 & & \text { in } \Omega \\ \phi &=0 & & \...
3
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0answers
75 views

Solving PDEs: What is the best way to deal with non-banded/dense jacobians?

I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
0
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1answer
138 views

Help me choose a book on the numerical integration of PDEs

For ODEs I have these books: Griffiths, David, Higham, Desmond J., Numerical Methods for Ordinary Differential Equations, 2010 Alfio Quarteroni, Riccardo Sacco, Fausto Saleri, Numerical Mathematics, ...
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3answers
275 views

How to begin writing scientific codes in C++ in Trilinos or PETSC style?

My background: I have taken some courses on numerical analysis during my PhD and read a few books on the topic as well. I mostly work on low Reynolds number fluid mechanics and use boundary element ...
8
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2answers
13k views

How to discretize the advection equation using the Crank-Nicolson method?

The advection equation needs to be discretized in order to be used for the Crank-Nicolson method. Can someone show me how to do that?
5
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1answer
250 views

What is the weak form of a vector type Laplace equation?

For a scalar variable $u$, the weak form of the following Laplacian: $$\nabla^2 u =0 $$ with the assumption that $v$ vanishes at the boundary is $$\int \nabla v . \nabla u \, d\Omega = 0 $$ which is ...
3
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3answers
263 views

What numerical methods are used to model deformations in elastic physics?

What numerical methods are used to model deformations in elastic physics? For example, here's an example of a hyperelastic deformation in Ansys: Perhaps more simply than hyperelasticity, for linear ...
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2answers
85 views

Stability of the Forward-Time Central Space method, section 9.3 in LeVeque

I am reading section 9.3 in Leveque about stability. The explanations are very short and brief so I am asking for some help to understand and elaborations. The result of the section is that for FTCS ...
2
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1answer
141 views

How to discretize a non-linear PDE with boundary conditions and intial value

Consider this non linear PDE: $$u_t + c(u^2)_x =\alpha u_{xx} \text{ , } -1<x<1 , t>0 $$ with $$u(-1,t) = g_L(t) \text{ , } u(1,t) = g_R(t) \text{ and } u(x,0)=f(x) $$ where the 3 functions(...
2
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1answer
53 views

How to demonstrate the order of convergence of FTBS method for solving a hyperbolic PDE

consider the Purely hyperbolic model problem $$u_t+au_x=0$$ $$u(-1,t)=u(1,t) \text{ (periodic boundary)}$$ $$u(x,0)=f(x)$$ with $f(y)=\sin(2\pi y)$. Furthermore the exact solution is given by $u(x,t)=...
2
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3answers
181 views

Solving a PDE with 2 variables, with one variable whose derivative with respect to space is only known

I am trying to solve a PDE of the form $$\frac{\partial u}{\partial t} = D \frac{\partial^2u}{\partial x^2} + k\ \ \ (1)$$ where only $k$'s first derivative with respect to $x$ is known $$\frac{\...
0
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1answer
273 views

Elliptic PDE finite volume method with Dirichlet boundary condition

I want to discretize the following equation using a Finite Volume Method $$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$ I'm using Voronoi cells here: ...
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0answers
51 views

Stability plot of upward difference implicit time

I am analyzing the stability of 1D convection (advection) equation as shown in the picture. When I derive the equations as shown I want to get rid of the complex number. I`m asking if those stability ...
2
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1answer
283 views

Weak formulation for advection diffusion reaction

I need a check on the following exercise about weak formulations and finite elements. Consider the advection diffusion system $$ \begin{cases} -(\mu u')' + \beta u' + \gamma u = f \\ u(a)=0 \\ u(b) = ...
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0answers
117 views

Error $L_{2}$ convergence in Finite Element for Poisson Equation

I have written a Matlab code to solve the equation $-u'' = f$ with conditions $u(0) = u'(1) = 0$ on the domain $x \in [0,1]$. I tested the code with $f(x) = -2, \forall x\in [0,1]$. I check the plot ...
3
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1answer
156 views

Finite element method for high-frequency electromagnetics

I am writing a project about the Finite element method for use in high-frequency solutions of Maxwell's equations. This could be for use in antenna design and similar. I have some trouble ...
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0answers
42 views

discretization of advection diffusion with variable coefficients

I am looking for help to find a somewhat stable FD numerical scheme for the advection diffusion equation posed on a curve $(x(r),y(r))$. The equation becomes $$u_t=\alpha(r) u_r +\beta(r) u_{rr}+f(r,t)...
0
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1answer
81 views

High Running Time and Suboptimal Accuracy of 2D Wave Equation Solver with Finite Differences

Im trying to solve the following 2D wave equation: $$u_{tt} = u_{xx} + u_{yy}, \hspace{3mm} u(x,y,0) = \cos(4 \pi x) \sin(4 \pi y), \hspace{3mm} u_t(x,y,0) = 0$$ with the periodic boundary condition ...
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1answer
155 views

Trouble Implementing 1d Wave Equation Finite Difference Solver

Im trying to solve the 1d Wave Equation on $x \in \mathbb{R}, t > 0$: $$u_{tt} = c^2u_{xx}, \hspace{5mm} u(x,0) = \cos(4 \pi x), \hspace{5mm} u_t(x,0) = 0$$ with $c = 1$ and a periodic boundary ...
2
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2answers
1k 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
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0answers
67 views

Divergence on wave equation simulation

I'm currenly working on my own PDE solver for non-linear simulations in python. I've done succesfully simulations for KdV and Fisher's equation, but now I'm playing with second order derivatives in ...
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0answers
60 views

Solving Laplace equation with constraint on boundary

I have found the following PDE problem in a paper: Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
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0answers
47 views

Cauchy problem ill-posed?

Find the solution to the Cauchy problem consisting of the wave equation : $$u_{xx}-u_{yy}=0$$ together with initial conditions: $$ u(x,0)=0,$$ $$u_{y}(x,0)=g(x)$$ for some known initial datum $g$. Is ...
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0answers
40 views

Output solution vector in Petsc

I am using petsc to solve a linear elasticity problem discretized by finite elements.The initial mesh is read by a mesh file and the distribution in each processor is done using METIS.I am using only ...

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