# 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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### Reference request for finite elements theory

Consider a domain $\Omega \subseteq \mathbb{R}^{2,3}$ which is non convex and with $C^2$ boundary. Could you recommend a good reference where it is explained how: without needing isoparametric ...
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### What condition ensures the global continuity of the solution in the FEM?

I know this is trivial but I don't seem to understand it. In which step of the FE formulation do we enforce the global continuity of the solution? Or in other words, how the construction of the local ...
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### Setting up boundary conditions to solve PDEs using method of lines

Objective: To add boundary/initial conditions (BCs/ICs) to a system of ODEs I have used the method of lines to convert a system of PDEs into a system of ODEs. The ODEs themselves involve a lot of ...
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### How to solve spatially discretised PDEs (method of lines) in solve_ivp or ODEint?

I can discretise the spatial domain of a system of PDEs using the method of lines, converting the system of PDEs to a system of ODEs (with a time derivative only). These equations (for context they ...
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### Direct integration of 2D Euler Equations with Runge Kutta shows oscillating Courant-Friedrichs-Lewy coefficient. Stiff or Bug?

By writing the direct integration of the 2D Euler Equations in a wide and short box where the fluid enters and exits through the horizontal faces using the Runge Kutta O(4) method I have found that ...
1 vote
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### 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 ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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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 votes 1 answer 82 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 votes 1 answer 219 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 ... 3 votes 1 answer 98 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 ... 0 votes 0 answers 53 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 ... 0 votes 1 answer 37 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 ... 1 vote 0 answers 52 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: $$\frac{\partial \vec{r}}{\partial \psi} (\psi, \zeta) ... 0 votes 0 answers 44 views ### introduction to non-overlapping domain decomposition in 1D I am new to domain decomposition and have been searching in google for an introduction in 1D which goes over the complete procedure from the continuous to the discrete problem as well as the algorithm ... 0 votes 0 answers 55 views ### Discretizing Multi-species Ion Exchange Equations by Finite Volume Method I'm solving a system of multispecies ion exchange equations (diffusion+drift fluxes) in 1-d spherical domain using finite volume method to obtain the ion concentrations at the next time step. After ... 1 vote 2 answers 181 views ### Continuation of solution to -\nabla\cdot (k(x,y)\nabla u)=f I'm trying to solve the following problem, I had previously opened another discussion for the implementation and well, it seems that it has turned out well, it can be found here. I need to calculate ... 3 votes 0 answers 88 views ### Strange Picard iteration I am interested in solving the equation$$ \begin{aligned} \nabla \cdot\left(\nabla \phi-\frac{\nabla \phi}{|\nabla \phi|}\right) &=0 & & \text { in } \Omega \\ \phi &=0 & & \... 3 votes 0 answers 82 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 votes 2 answers 372 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 ... 0 votes 1 answer 159 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, ... 5 votes 1 answer 378 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 ...
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(...