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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Libraries with the method of lines for parabolic PDEs

Could you please advise some programs or libraries for solving parabolic PDEs (or its systems) in 1D, 2D and 3D, for example, with the method of lines? The system of parabolic PDEs can be nonlinear in ...
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60 views

How to determine the initial values of compressible Euler equations for shock wave problem by knowing initial Mach number?

One of the famous and convenient test case for shock wave problem is Sod's Shock tube test case for 1D PDEs. The initial set up has been done by dividing domain half with diaphragm. Left and right ...
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88 views

Numerical method of lines for solving PDEs

Could you please advise some literature about the numerical method of lines (MOL) for parabolic PDEs? It is a method of solving PDEs with discretizing only by space but not by time. A system of ODEs ...
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68 views

Solving Advection (Convection) - Diffusion - Reaction Partial Differential Equation in Python

I am looking for library written in Python which will enable me to solve the coupled nonlinear equations which looks like: I need the library which will enable me to couple this solver to other ...
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What is the exact formulation of compressible Euler equation of gas dynamics in polar coordinates with artificial diffusion in 2D?

The interested equation is advection-diffusion equation. One of the canonical example is Navier-Stokes equations. However, I would like to let the coefficient of diffusion constant goes to zero, ...
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60 views

Looking for some literature about numerical solution of _coupled_ PDEs

There's a vast amount of literature about the numerical solution of partial differential equations. But in none of my books or lectures did I find anything about coupled systems. Internet and library ...
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Wave Equation with Constant Boundary Conditions [migrated]

I need to find a formal solution to \begin{eqnarray} &u_{tt} &= c^2 u+{xx}, \;\;\;0<x<1, \mathrm{and \;}t>0\\ &u(x,0)&=x+1,\\ &u_t(x,0)&=x(1-x), \;\;\;\;0 \leq x \leq ...
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93 views

steady state solution from parabolic problem vs solution of elliptic problem

My question is related, but not a duplicate of Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde Suppose I solve the parabolic PDE: $u_t = \Delta u + ...
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151 views

Implementation of nonlinear term in FEM

Although there are similar questions, I am also struggling with the implementation of the following term in "my own code" by Finite Element Method, namely, $\nabla \phi \cdot \nabla \phi$. $\phi$ is ...
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1answer
77 views

Numerically solve a PDE in Python with a term calculated by coarse-graining

I'm trying to solve a PDE in Python of the form, $\dfrac{\partial c(\mathbf{x}, t)}{\partial t} = \mathrm{D} \nabla^2 c(\mathbf{x}, t) -\gamma \rho(\mathbf{x}, t) c(\mathbf{x}, t)$ where $c$ ...
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90 views

Partial differential equations with octave [closed]

I need to find a numerical solution for $-\Delta U = f$, on the $\Omega = [0,1]^2$, with $ U|_{\partial \Omega} = 0$. I found a method: POISSONFD in ...
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94 views

Is there a jump condition for this PDE? ( Brinkman model , piecewise constant permeability)

The Brinkman equations for steady flow of an incompressible fluid through rigid porous solid are: $-\dfrac{\mu_0}{k}\mathbf{v} - \mathrm{grad}p + \mu_0 \Delta \mathbf{v} =0$ and ...
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1answer
84 views

How to solve coupled steady laminar diffusion flame jet problem? [closed]

I am trying to solve governing equations of laminar diffusion flame jet for steady state case. In the next step, I will solve for unsteady case. I have non-dimensional continuity, axial momentum, ...
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2answers
197 views

Visualising Maxwell's equations using MATLAB

I asked this question to help me understand what is going on in one of Maxwell's equations. I am happy with following through the maths on paper now, but would like to use MATLAB to take it one step ...
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121 views

Finite differences and Neumann boundary conditions

I am dealing with a highly nonlinear system of two PDEs. I already have a code to solve the system in case of Dirichlet boundary conditions. The explicit system is: $$ \begin{eqnarray*} \partial_{t}u ...
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217 views

Understanding the cost of adjoint method for pde-constrained optimization

I'm trying to understand how the adjoint-based optimization method works for a PDE constrained optimization. Particularly, I'm trying to understand why the adjoint method is more efficient for ...
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2answers
91 views

use Matlab's PDE toolbox to solve PDE with variable coefficients [closed]

I'm new to the PDE toolbox in Matlab. From the PDE specification window of the toolbox, it looks like one can only solve PDE with constant coefficients. How can ...
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198 views

Books on mathematical foundation of finite element methods

After reading three books about finite element method, with two of them covering also finite volume and grid generation, I found myself lost when I have to discuss these topics with library developers ...
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136 views

Finite Elements Weak Formulation generalization

I am struggling with an equation that represents the Weak form of Galerkin method: $ \phi^{T}F(\textbf{u})\sim \int_{\Omega}^{ } \phi.f_{0}(\mathit{u},\nabla \mathit{u}) + ...
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128 views

Finite element convergence rates for mixed problems

I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh. ...
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153 views

vector PDEs on manifolds

What are the subtleties involved in solving vector PDEs on manifolds? Can someone suggest a reference summarizing the problems involved? Specifically I want to solve a vector Helmholtz equation with ...
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161 views

wavelet for numerical partial differential equations

Is there a good introduction into wavelet Galerkin schemes for numerical partial (and ordinary) differential equations?
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136 views

How to compute $\Delta u$ on the boundary of the biharmonic equation?

Let $u$ be the answer of a PDE.Is there any relationship between $u,\frac{\partial u} {\partial n}$ and $\Delta u$. I have the values of $u$ and $\frac{\partial u} {\partial n}$ on $\partial \Omega$ ...
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101 views

Time Integration of a nonlinear reaction-diffusion system

I want to solve the following system of nonlinear reaction-diffusion equations (Schnakenberg Turing) using FEM methods (such as deal.ii): $$ \partial_{t} u = \Delta u + \gamma\left(a-u+u²v\right)$$ ...
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28 views

Second order PDE with minimum condition

I need to solve 2nd order PDE of $F=F(X,Y)$, where X and Y are functions of t. I have got to point where I transformed my PDE to canonical with logarithmical transformation for X and Y to get constant ...
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How should I choose the knot sequence when using B-splines as a basis for solving a PDE?

I'm looking to solve the Schrödinger equation with a basis made of a tensor product of basis splines. A number of papers describe calculations made with a program designed this way, but they never ...
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57 views

Guidelines for choosing manufactured solutions for numerical PDE schemes

When testing a numerical method for a PDE, I know that it's often useful to compare it to a known analytical solution. If none is available, one can always 'manufacture' a solution, substitute it ...
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432 views

What are the relative benefits of using Adams-Moulton over Adams-Bashforth algorithm?

I am solving a system of two coupled PDE's in two spatial dimensions and in time computationally. Since the function evaluations are expensive, I would like to use a multistep method (initialised ...
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2answers
133 views

biharmonic equation

I want to solve the biharmonic equation numerically, that is: $$\Delta^2 u=f~~in~~\Omega$$ $$u=g_1~~on ~~\partial \Omega$$ $$\frac {\partial u}{\partial n}=g_2~~on ~~\partial \Omega$$ Using Green's ...
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62 views

Performance metrics to compare initial-boundary value problem solutions

I am comparing the performance several finite difference methods of solving an initial-boundary value problem. There are several dimensions to this comparison: Number of cells Number of timesteps ...
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108 views

Integration of nonlinear PIDE via spectral methods

At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction ...
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259 views

Coupled nonlinear PDEs with time dependence on the RHS

I would like to numerically solve the following system of 2 coupled partial differential equations for the unknown functions $\psi_X(x,y,t)$ and $\psi_C(x,y,t)$: $\partial_t \psi_X = -i\psi_C - ...
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105 views

Solving the quadratic in the Fast Marching Method

The Fast Marching Method is a way of solving the Eikonal Equation on a discrete grid, essentially just computing a wavefront speading out from initial points, IE: The idea is that we want to ...
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133 views

Solving a system of nonlinear PDEs by minimization

I have two coupled nonlinear partial differential equations of the form: $ \begin{align} \dot{u} -f(u,u',u'',v,v',v'')=0 \\ \dot{v} -g(u,u',u'',v,v',v'')=0 \end{align} $ The boundary conditions are ...
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A Question About Weak Forms in Fenics

Is it possible to use test and trial functions from two different function spaces (defined over two different meshes) in a single weak form? Under what conditions can I do this (eg., each term in the ...
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3answers
157 views

Surface normals integration

I am trying to reconstruct a 3D surface given the normals of the unknown surface. Reading through this paper on section 4 they say [...] denote the surface by $z(x,y)$. The directions of the ...
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1answer
85 views

Numerically solving the polar poisson equation [closed]

I want to solve the Poisson equation for a 2D polar system: $$\Delta_r f(r) = u(r)$$ with the Laplace operator: $ \Delta_r f(r) = \frac{1}{r}\frac{\partial}{\partial r} \left[r ...
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1answer
62 views

Imposing symmetry plane boundary condtition

I want to impose symmetry plane boundary condition for a solid mechanics problem. I googled around and found out that in many places people say to "forbid displacemnts out of symmentry plane and ...
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Segregated solving of a tightly coupled system of PDEs

To compute the evolution of a free surface between two incompressible, immiscible liquids, two tightly coupled equations have to be solved, the volume fraction advection and the Navier-Stokes ...
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Strategy for solving a non-trivial differential equation

I would like to numerically solve an equation of a type as shown below. Does anyone of you have an idea how to approach such a problem? Any links to literature or for further reading would be greatly ...
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1answer
247 views

Parallel 2d heat equation (implicit timestepping) using MPI

I am trying to solve the time dependent heat equation with backward euler timestepping and second order space finite differences. This results in a Poisson system needing to be inverted. In serial ...
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1answer
125 views

Von Neumann stability analysis in 3d

I need to get a stability criterion for the numerical scheme for equation $$\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}-\frac{\partial^2 ...
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1answer
120 views

Courant Friedrichs Lewy condition - how to get it?

I am interested, how can we get CFL condition for every type of PDE? It's known that for 1st order linear equation $$\frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}=0 $$ CFL is get from ...
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127 views

Derivation of a Higher Order Compact Alternating Direction Implicit Method

I dont understand how this Higher Order Compact ADI scheme, which is fourth order in time and space, for the wave equation is derived: I go through the following Using Tylor's expansion $u(t+h,x,y)$ ...
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62 views

Locally conservative method for differential generalized eigenvalue problem

I have to approximate the smallest eigenvalue of the following generalized eigenvalue problem $$ - \nabla \cdot D(x) \nabla p(x) + \alpha(x) p(x) = \lambda \beta(x) p(x) $$ over a domain like ...
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109 views

Discretization method for a reaction dominated elliptic PDE

I'm working with an elliptic reaction diffusion PDE of the form $$-k\nabla^2u+cu=0$$ I've noticed that when the reaction term dominates over the diffusion (i.e. $c>>k$), the true (exact) ...
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The effect of decoupling a coupled system of PDEs

I asked a somewhat similar question previously but perhaps it might have been too specific for anyone to really answer. Here is a bit more general of a question that I am struggling with. Consider the ...
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1answer
151 views

User Friendly GUI based 3d PDE solving software

I'm searching for software which can solve 3d IBVP like PDEtool in Matlab (user-friendly GUI, easy to learn how to use). What could you advise to me? (ANSYS,COMSOL,OpenFOAM are too difficult for me)
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162 views

Spectral Methods in time

I was reading up on Spectral Methods for PDEs. In all the descriptions I read, while the position component is approximated via a Fourier series or other methods, the time component is still ...
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118 views

Sequential approach to solving coupled PDEs

I'm dealing with a coupled system of three transient, non-linear convection-diffusion equations. Let's just say to simplify the problem that they take the following form: $$ ...