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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.

3
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2answers
123 views

citations for numerical lookup table interpolation of P/ODE(s) RHS

I'm not sure that this *overflow is right place to ask.... Sorry if it is off topic. Does anyone know a citation (scientific article or book) for a numerical trick (method), when we tabulate a right-...
1
vote
1answer
207 views

How can I numericaly solve a convection-diffusion equation with a large diffusion term?

I want to numerically solve the advection-diffusion equation: \begin{equation} u_t(x,t) + cu_x(x,t) = v u_{xx}(x,t) \end{equation} for $x \in [0,1]$ and $t \geq 0$ subject to the boundary conditions ...
4
votes
1answer
117 views

Stable method for solving a HJB equation

I am considering finite difference methods and their error analysis for solving HJB equation of the following form: $$ v_t=|\sigma(x)v_x|,\quad x\in \mathbb{R}, $$ where $\sigma$ is a given function ...
3
votes
0answers
113 views

How one could choose the value of viscous coefficient for obtaining stable solution of Burgers' equation?

Burgers' equation is a fundamental PDE used in various fields such as number theory, gas dynamics, heat conduction, elasticity, etc. It is crucial especially for developing numerical models for ...
1
vote
0answers
44 views

Degree of freedom for elastic wave propagation problem

I am solving a elastodynamics (vector valued elastic wave) equation. I create the 2D mesh in Gmsh discretised into triangular elements of second order. Therefore, it is my understanding that the ...
1
vote
0answers
87 views

Which numerical scheme should be used?

Trying to find a way to solve exponentially nonlinear elliptic equation with complex source term i manage to know about such schemes like Godunov, Lax-Friedrich, MUSCL. I still seraching for ...
3
votes
1answer
292 views

On the Rellich-Kondrachov embedding theorem

Let $\Omega$ be a bounded open set in $\mathbb{R}^d$ where $d \geq 1$ is a positive integer, with Lipschitz boundary. Let $k,l$ be non-negative integers and $1 \leq p < \infty $ then if $k > l$ ...
1
vote
2answers
262 views

implementation of method of line and Runge-Kutta to the given equation

$$\frac{d^2 u}{dx^2} +A \frac{d^2}{dx^2}\left(\frac{du}{dt}\right)=B $$ I want to solve the equation given above. I need to first discretize it by the Method of Lines and then evolve the resulting ...
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0answers
135 views

Numerical solution of non-linear heat-diffusion PDE using the Crank-Nicholson 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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0answers
56 views

Determine Lagrange nodal variables of a simplex $T$

Consider a simplex $T$ in $R^d$ with $N_1(T) = \left\{N_i\right\}_{i=0}^{d}\subset P_1^{*}(T)$ be the Lagrange nodal variables (or nodal evaluation). By the Riesz representation theorem, there exist ...
8
votes
1answer
113 views

reformulating inverse problem as multi-objective optimization

I'm working on an inverse problem for my Ph.D. research, for which I'll write the objective functional as $J(\theta) = E(G(\theta) - u^o)$, where $\theta$ are the parameters, $G$ is the forward map ...
0
votes
1answer
106 views

Role of non-hermitian coefficient matrices in the discretization of self-adjoint operators

What is the role of non-symmetric coefficient matrices in the solution of partial differential equations with self-adjoint operators? Particularly, I'm thinking about time-propagation of a linear ...
0
votes
1answer
460 views

Heat equation with Neumann and Dirichlet conditions on same boundary

I am looking at numerical solutions to the heat equation with Dirichlet and Neumann conditions on the same boundary. That is $u(x,t)$ satisfying $$ u_t = u_{xx}\,, \quad x \in[0,1]\,, \quad t>0\,,...
1
vote
4answers
317 views

Numerical solution to 2D divergence equation

Is there any way to numerically solve the following two-dimensional equation: \begin{equation} \nabla_{xy} \cdot \vec{f}(x,y) = a(x,y) \end{equation} on a rectangular grid, knowing that $\vec{f}(x,y)$ ...
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vote
0answers
74 views

Implementing initial conditions into the solution domain of a 1-D advection-diffusion equation

I have the following PDE. $\frac{\partial c}{\partial t} + v\frac{\partial c}{\partial x} - D\frac{\partial^2 c}{\partial x^2} = 0 \\$. I have discretized it such that i now have $\frac{dC}{dt} = ...
1
vote
0answers
216 views

Algorithm for face based data-structure - CFD

Good morning I'm trying to develop an unstructured CFD code to solve Euler equations in a finite-volume (cell-centered) context (learning purposes). I was able to build from a cgns file some basic ...
1
vote
2answers
236 views

Solving PDE in 1D with FD and MATLAB

I have to solve the following equation: $-u_{xx}=1$, with $x\in(0,1)$ and $u(0)=0,u(1)=0$. I have to solve it with the following numerical scheme: $\frac{1}{h_k^2}(-\frac{1}{2}u_{k-1}+u_k-\frac{1}{...
2
votes
2answers
2k views

Understanding the Courant–Friedrichs–Lewy condition

I understand these equations in particular can be solved easily without use of computational methods. Although right now I am concerned with trying to solve these equations using numerical integration ...
1
vote
1answer
82 views

Classification of method for solving PDEs

If I have a system of equations as follows (where $i = \sqrt{-1}$): $$ \frac {\partial A}{\partial t} = iA^*B - A \tag{1} \\ $$ $$ \frac {\partial B}{\partial z} = AB^* - B \tag{2} $$ Using the ...
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0answers
121 views

Using backward difference approximations for higher order derivatives

I am trying to solve a system of equations and have a question regarding the validity of my approach when implementing a fifth-order Cash-Karp Runge-Kutta (CKRK) embedded method with the method of ...
1
vote
2answers
190 views

derivative of an array

Hi I am trying to take a derivative of an array but am having trouble. The array is two dimensional, $x$ and $y$ directions. I would like to take a derivative along $x$ and along $y$ using central ...
1
vote
0answers
35 views

Solving HJ using Hopf-Lax formula (Burgers case)

Hi I want to approximate $u(x,t)$ solution of: $\begin{cases}u_t+H(u_x)=0 \\u(x,0)=u_0(x)= e^{−(x−3)^2} \end{cases}$, where $H(u_x)=\frac{u_x^2}{2}$ and my spatial domain is $\Omega=[0;10]$. Now i ...
1
vote
1answer
285 views

Solving first versus second order PDE

I am trying to numerically solve a PDE, and just had a question as to the validity of a certain approach. For example, given the PDE: $$ \frac {\partial ^2 E}{\partial t^2} = - k\frac {\partial E}{\...
0
votes
3answers
106 views

Choice of solver/software for global optimisation of cheap black-box function with known derivatives

I am trying to estimate a few unknown parameters of my continuous non-linear PDAE model (simulated through finite-volume method spatial discretisation, and time-stepping through method-of-lines). I am ...
1
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1answer
82 views

3 questions on FEM to solve elliptic PDE with homogeneous and mixed boundary conditions

Assume we use FEM with piecewise linear finite elements to discretize the BVP over $\omega = (0,1)$: $-u''+ bu' + u = 2x$, $u(0) = u(1) = 0$ for parameter $b\in R$. Given a mesh $T = \left\{x_i\right\}...
0
votes
1answer
274 views

Implementing odespy for system of PDEs

After trying to use RK4 to solve the below system of equations, it appears the output had large and fast oscillations even with an adaptive time step I incorporated using the Cash-Karp method. I am ...
1
vote
0answers
76 views

What is a good algorithm to solve a discrete continuity equation in Cylindrical coordinates?

The equation is: $\partial f/\partial t + \nabla \cdot (v f) = 0$ $, \;\; f \in [0,1] $ and $v$ is a velocity known at every grid cell. A more precise constraint is that $f$ is either 0 or 1 but ...
0
votes
1answer
199 views

Applying Runge-Kutta to nonlinear system of PDEs

I am applying a 4th order Runge-Kutta code, using the method of lines, to solve the following: \begin{equation} \frac {\partial y_1}{\partial t} = y_2 y_3 - C_1 y_1 \end{equation} \begin{equation} \...
0
votes
2answers
272 views

Stable implicit method to solve convection-heat diffusion in 3D

The last couple of hours I have been looking for an unconditionally stable method to solve the convection-diffusion equation within a 3D inhomogeneous material. Here's the well known diffusion-...
0
votes
1answer
141 views

How to compute matrix representation of $\hat{y}\frac{\partial}{\partial x}$?

I have a 2-dimensional system which I would like to solve numerically (I'm using finite difference method right now), and its an eigenvalue problem. I have a term that looks like $H\psi(x,y) = [-\frac{...
0
votes
1answer
66 views

Difference Equation PDE in MatLab

I would like to program the following difference equation. Find numbers $v_{i,j}$ so that for $1\leq i\leq 4$, $0\leq j\leq 5$: $$ v_{i,j+1} = (0.1)v_{i+1,j} + (0.8)v_{i,j} + (0.1)v_{i-1,j} $$ In ...
3
votes
0answers
124 views

Multigrid for Robin boundary conditions

I have a question regarding the treatment of Robin boundary conditions in a multigrid solver. I am solving the Poisson equation in $\Omega=(0,1)^2$ with Robin boundary conditions on the boundary, $$- \...
1
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0answers
156 views

Algorithm for Adaptive Mesh Refinement

I am trying to implement Adaptive Mesh Refinement. I am not a Mathematics/Computational Science person so I will try to write the algorithm in a simpler way. I will be grateful if experts can comment ...
0
votes
0answers
37 views

Useful Quantity for Heat Equation? [duplicate]

I'm interested in testing some algorithms on the heat equation, and I'd like to assess their accuracy. When evolving a Hamiltonian system, one has the energy to check the validity/correctness of the ...
1
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0answers
60 views

Dielectric in an external electric field

I want to numerically solve for electric field of an object in an external electric field. The object has linear displacement field such that its permitivity ($\epsilon$) is constant. The object's ...
3
votes
2answers
154 views

Gradient computation in AMR framework (Green-Gauss theorem)

I'm currently working with this kind of mesh (AMR) : To compute the gradient at the center of the cell P, I use the following formula : $$\nabla \phi_P = \frac{1}{\Omega} \sum_{faces} \phi_f \vec{S_f}...
1
vote
0answers
140 views

Numerically solving a system of stiff nonlinear PDEs

I am attempting to numerically solve the following: \begin{align} \frac {\partial y_1}{\partial t} &= i(y_2y_3 - y_2^*y_3^*) - y_1 \tag{1}\\ \frac {\partial y_2}{\partial t} &= y_1^*y_3 - y_2 ...
4
votes
0answers
117 views

Eigenvalues of a Laplacian operator on an irregular mesh

I have the following setting: An irregularly-shaped domain, expressed as a mesh of points A Laplacian operator, together with boundary conditions I am looking for the eigenvalues of that operator, i....
0
votes
1answer
112 views

Imposing boundary conditions for PDE quadratic eigenvalue problem

I have a quadratic eigenvalue problem of the form: $$(A_2 s^2 + A_1 s + A_0)\hat{v} = 0$$ where $s$ is the eigenvalue. The matrices $A_i$ contain derivatives up to order six, and I have six boundary ...
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0answers
59 views

Name of third-party Matlab packages for solving PDEs

I am interested in working with systems of PDEs in two spatial dimensions, as part of a mathematical modeling project. The systems of PDEs are coupled, and there is no convenient reduction to 1 ...
1
vote
1answer
247 views

FD implementation of Absorbing Boundary condition for acoustic wave

I am simulating acoustic wave equation in which absorbing boundary condition has to be applied. It is applied in two ways Ist method is as mentioned in this paper. Boundary condition at bottom is (...
3
votes
1answer
177 views

Numerical methods for coupled stiff PDEs

I'm dealing with a set of nonlinear coupled PDEs that have the form: \begin{align} \frac {\partial y_1}{\partial t} &= y_2y_3 - y_1 \tag{1}\\ \frac {\partial y_2}{\partial t} &= y_1y_3 - y_2 \...
0
votes
2answers
237 views

How to choose a simple manufactured solution for Euler equation?

In order to verify a two-phase subsonic compressible isothermal Euler code, I am trying to implement a manufactured solution following what is discussed here and references therein. Also, as an ...
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0answers
53 views

Direction-splitting for SSP-RK schemes

What are the implications of applying a direction-splitting within each stage of an SSP-RK scheme? For instance, given a standard advective transport type equation: $$ \partial_{t}Q + \operatorname{...
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0answers
89 views

Numerical method to find stationary solution of an PDE in the phase space

I'm relatively new in numerical methods for PDEs, so I have a question about solving the following PDE numerically for $W(x,p)$ in the phase space: $$-p\frac{dW}{dx}+x\frac{dW}{dp}+\frac{d}{dx}\left(...
1
vote
1answer
113 views

Stability condition for explicit/implicit via non negative coefficients

To make stability proofs simpler, I can consider an explicit scheme written as $$V(n+1,i)=aV(n,i-1)+bV(n,i)+cV(n,i+1)$$ and one can show that if $a,b,c\ge 0$ and $a+b+c\le1$, then the explicit method ...
2
votes
1answer
79 views

High order time splitting methods

There are lots of higher order time splitting method as shown by the list with real and complex coefficients $a_i, b_i, c_i$: $$ [e^{c_s \Delta t \hat C}] e^{b_s \Delta t \hat B} e^{a_s \Delta t \...
1
vote
2answers
551 views

Strang splitting

I have recently come across the Strang splitting and have some questions. For the differential equation of the form $$ dy/dt = (L_1 + L_2)y$$ Strang splitting implement the time splitting as $$ \...
1
vote
0answers
59 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 &...
3
votes
1answer
142 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 ...