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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
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
97 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 ...
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2answers
100 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
128 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 ...
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1answer
271 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
128 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
116 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
205 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
146 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
73 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
99 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
334 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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2answers
190 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
68 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
110 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
253 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, ...
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0answers
33 views

How do 'virtual kinematics/functions' play a role during deriving weak form formulations for physical problems?

I wanna ask a question that confuses me quite a long time. I saw many guys, in the context of computational mechanics, they seemed to choose the virtual functions or kinematics in a way that some ...
3
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0answers
81 views

What is Chebfun `eigs` doing

What is this doing? Looks like the original eigenvalue problem is converted into generalized eigenvalue problems with different dimensions of collocation points. Can someone explain more about this? ...
0
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1answer
76 views

How to treat non-linear term in finite difference solution of $T''_x+T''_y+aT^2=0$?

Can we linearize $T^2$ When solving $T''_x+T''_y+aT^2=0$ by finite difference? I solved $T''_x+T''_y=0$ in Matlab using a finite difference explicit scheme. But when there is a source term, I come ...
1
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1answer
184 views

proper derivation of a functional for a time dependent parameter estimation problem

Following my previous question and its answer, after some reading of the advised books, I'm still confused about how to get the derivative of the functional to find the best parameter of my reaction ...
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0answers
113 views

Reference request: free/open source books on intro numerical PDE

I'm interested in finding a free online textbook that is an introductory textbook on numerical PDE for engineers. It should cover the basics of finite differences, finite elements, and Fourier series ...
4
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1answer
227 views

adjoint method for reaction-diffusion problem

I'm trying to code a parameter estimation for a reaction-diffusion problem. Namely, knowing the distribution of tumor density $u$ at time $0$ and $T_f$ ($u^0$ and $u^f$), what are the best ...
0
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1answer
115 views

pde-constrained optimization

I'm trying to solve a problem where I have a initial and final distribution of tumor, and my goal is to find the best map of parameters (diffusion and reaction terms) for a reaction-diffusion equation,...
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0answers
39 views

Understanding “characteristics”: texts/references and background needed?

I'm trying to get on top of "characteristics" as a way of solving fluid dynamic PDEs, coming from this question and using the textbook suggested there (Whitham, Linear And Non-linear Waves). I'm ...
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0answers
83 views

FiPy with derivative source terms

I have a coupled nonlinear PDE system in 1 spatial dimension, in which I want to solve using FiPy. The dependent variables are $n$ and $T$: \begin{align} \frac{\partial n}{\partial t} \,&=\, D\,\...
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1answer
173 views

Is there a general analytic solution to 1D advection of velocity, $u_t=-uu_x$?

This is to help me relate continuous and discrete, predict what my scheme should be doing, and move toward using the method of manufactured solutions. There's a solution for constant velocity $c$ ...
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2answers
102 views

Computable alternative to “almost everywhere”

I am working with finite elements for Maxwell's Equations (i.e. with Nedelec's edge elements) and for computation I'm using the FEniCS-project. While implementing the Augmented Lagragian Method, I ...
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0answers
45 views

Finding minimum of Wronskian determinant

If I have a partial differentail equation such as $\frac{\partial^2 u}{\partial t^2} + c^2 \frac{\partial^2 u}{\partial x^2}$, with boundary conditions $u(0,t) = 0$ and $u(1,t)=0$, I can solve this ...
2
votes
1answer
124 views

Galerkin method for a system of nonlinear PDEs

Suppose I have a nonlinear system of PDEs. I am actually interested in Navier-Stokes, but, for the sake of simplicity and example, suppose I had $$ \frac{\partial f}{\partial t} - f \frac{\partial g}{...
2
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0answers
35 views

One dimensional differential equation with resonance recombination (RHS)

I am trying to solve the following equation: $$\dfrac{\partial }{\partial \epsilon}\left[B\left(\epsilon\right)f_e\left(1-f_e\right)+D\left(\epsilon\right)\dfrac{\partial f_e}{\partial \epsilon}\...
3
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0answers
81 views

Piecewise Constant Enrichment of the continuous galerkin method

I am interested to study crack propagation in a hyperelastic material in a variational setting. The crack surface exhibits a jump discontinuity. The function space for displacement field should ...
3
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1answer
95 views

Discontinuous Galerkin FEM : Control points are mid-points of edges instead of nodes

I am thinking to use discontinuous galerkin FEM (DGFEM) method to estimate discontinuous displacement field $u: \Omega \rightarrow \mathbb{R}^2$ at the crack surface of a material. The domain is ...
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0answers
89 views

Hello word in FEniCS? [closed]

I am trying to start using FEniCS, but have a problem with the simple hello world examples given in the books. Could you please give me the simplest hello world ...
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0answers
79 views

Code for solving the heat equation on the semi-infinite rod

Cross posted in mathematica.SE. Question : I want to test the solution which is given below is right by Matlab/Maple/Mathematica. Please look the post in mathstackexhange or Please look below. ...
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2answers
121 views

Learning differential equations: a textbook

I am a Computational Biology master's student and I would like to study differential equations (both ODEs and PDEs). I tried to have a look at the available textbooks but what I found is either too ...
6
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4answers
508 views

Why do I still obtain a unique solution with one-sided formula when b.c. isn't enough?

Let me illustrate the issue with an simplified example. Suppose we want to solve the following problem with finite difference method (FDM): $$\frac{\partial u(t,x)}{\partial t}=\frac{\partial^2 u(t,x)...
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0answers
43 views

Appropriately handling boundary conditions in a PDE eigenvalue problem

Suppose I have a nonlinear second order Cauchy PDE $\dfrac{\partial p(x,t)}{\partial t}=N(p(x,t))$, where $N:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$, and a known fixed point $u(x)$. Mathematically,...
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0answers
85 views

1D-Diffusion + chemical reactions: non-inear PDE-System with variable coefficients

I'm modeling a process which involves heat diffusion in 1D as well as different chemical reactions modifying the temperature. The system of PDEs looks something like this: $ \partial T/\partial t=-A\...
3
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0answers
184 views

Von Neumann analysis for coupled PDE's

I am interested in understanding how to perform stability analysis for coupled (to keep things do-able, lets say linear) PDE's In the case of a single PDE, i understand the logic behind the VN ...
4
votes
1answer
104 views

Bubnov-Galerkin method in 1D: how to handle convective-type nonlinearity?

Consider the BVP: find $u = u(x)$, for $x \in (0,1)$ that satisfies \begin{align} u'' + u u' = f, \\ u'(0) = g_n, u(1) = g_d. \end{align} To derive the weak form for this BVP, we multiply the first ...
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2answers
303 views

Newton method for a nonlinear system of time-independent PDEs

I have taken a course on undergrad scientific computing which discussed nonlinear algebraic equations about half-way in, and PDEs at the very end, but never discussed nonlinear PDEs. However in my ...
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1answer
78 views

implicit method (crank-Nicolson) I not understand the procedure [closed]

I'm trying to understand the passage through this equation can be written for easily solved with the fortran alghorithm in particular i don't understood the meaning of L_x and L_xx ... what (-1,0,1) ...
7
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1answer
202 views

Why is the continuous Galerkin Finite Element Method a poor choice for the inverse problem for the Navier-Lame equation?

I work in a field (elasticity reconstruction) that frequently uses standard Finite Element methods to solve the first order PDE, where we are given u and solve for mu and lambda: $$ (\mu(u_{i,j} + u_{...
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0answers
158 views

Solution to 1D consolidation problem python implementation

A solution to the 1D consolidation problem is given by $$\frac{\partial}{\partial t} p = c_{v} \frac{\partial^{2}}{\partial y^{2}} p$$ where $p$ is the pore water pressure, $c_v$ is the ...
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3answers
1k views

Galerkin method: Test functions vs. Basis functions

I'm a novice to finite element and I'm finding quite hard to find the actual difference between Test function(s) and Basis function(s). I would be glad if somone could explain me that and point out ...
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0answers
62 views

Solve a PDE BVP with Spectral methods in time and space?

I have a PDE (coming from a Bellman equation with $z$ under brownian motion). Let $z \in [0,\infty)$ and $t \in [0,T]$. To sketch the equation: $$ (r - g(t)) v(t,z) = \pi(t,z) + (\gamma - g(t))v_z(...
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1answer
240 views

How to implement finite difference method for one dimensional Navier-Stokes PDEs

I am trying to use backwards finite difference method to numerically solve a pair of partial differential equations: $\frac{\partial \left(pv\right)}{\partial x}+\frac{\partial p}{\partial t}=0$ $\...
2
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1answer
127 views

Finite difference method not working for advection PDE with negative coefficient

I'm trying to solve a very simple advection PDE $\frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x}=0$ where $c<0$. I have been able to implement a simple Modelica code to solve the ...
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0answers
63 views

BTCS-like method for heat conduction in unstructured triangular grid

I want to write a simple simulation for heat conduction in a unstructured triangular mesh. I already made it work for a structured rectangular grid with the ADI method, but now I need more complex ...