1
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1answer
28 views

Finite Difference Method Stability with diffusion equation

The diffusion equation is: $ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $ An explicit finite difference approach can be used to solve this, forward in ...
0
votes
0answers
20 views

Space-dependent diffusivity and finite-differences

I want to implement a finite difference code of this simple diffusion equation with space-dependent diffusivity: $$\partial_{t}u =D\partial_{x}^{2}u+\partial_{x}D\cdot\partial_{x}u$$ I go for a ...
2
votes
0answers
96 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
1
vote
0answers
19 views

heat equation with Interface Crank Nicolson

I am currently working on solving the heat equation with an interface numerically using Crank-Nicolson. There are jump discontinuities at the interface which are dealt with using fictitious values ...
1
vote
0answers
30 views

Stability conditions

Below is a problem about stability conditions that I have been struggling with it during an exam: Find the stability conditions for $$A\left ( \frac{\partial^2 u(x,\, y,\,t)}{\partial x^2} + ...
0
votes
0answers
27 views

Finite Differences and Scale Invariance

The semilinear heat equations $u_t = u_{xx} + u^p$ is invariant under the one parameter family of scalings $ t \to \lambda t, \quad x \to \lambda^{1/2}x, \quad u \to \lambda^{-1/(p-1)}u$. When we ...
0
votes
0answers
28 views

Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...
1
vote
1answer
87 views

In the numerical solution of the Wave Equation, using finite differences, where do I obtain the spatial values from?

In trying to implement a simplistic numerical solver for wave equations, I have run into a conceptual problem that I haven't been able to solve. Consider a one-dimensional wave equation of a quantity ...
0
votes
1answer
20 views

von Neumann stability analysis for irregular meshes

All the litterature I have come across about the von Neumann stability analysis is performned on regular grids. Can the analysis be performed analytically on irregular grids, or does it have to be ...
0
votes
2answers
152 views

Understand 1D FEM solution using quadratics elements

I'm a bit confused about applying the FEM using piecewise linear functions. I think I get understand how to use linear functions. We use the hat function for each element and the solution is ...
1
vote
1answer
183 views

Finite difference for variable conductivity

I'm trying to discretize a portion of the heat equation for a sphere and for a cylinder where: $r$ = radius, $T$ = temperature, and $k$ = thermal conductivity. for the cylinder shape: ...
3
votes
1answer
145 views

Discrete Bessel Functions

I am reading the paper "Discrete Bessel Functions" by R.H. Boyer (Journal of Mathematical Analysis and Applications. Vol 2, Issue 3, June 1961, pg. 509-524) and he begins the paper by discussing ...
3
votes
1answer
177 views

Stability of the BTCS scheme for the heat equation in a disk

Consider the $1$-D heat equation: $$ u_t = a \Delta u = au_{xx} \\ u(0,t) = u(1,t) = 0 \\ u(x,0) = u_0(x) $$ where $a > 0$ is constant and $u_0$ is given. It is a classic result that the implicit ...
6
votes
3answers
6k views

What is difference between Finite Different Method, Finite Element Method and Finite Volume Method for PDE?

Can you help me explain the basic difference between FDM, FEM and FVM? What is the best and why? Advantage and disadvantage of them?
2
votes
1answer
507 views

What is convection-dominated pde problems?

Can you explain for me what is convection-dominated problems? Definition and examples if possible. Why don't we can apply standard discretization methods (finite difference, finite element, finite ...
1
vote
1answer
600 views

Solving Laplace's equation using finite differences

I having coding in MatLab to approximate solutions to Laplace's equation in 2D using finite differences. I was able to do it without much problem. I learnt about how to implement this using this: ...
1
vote
1answer
120 views

Centered-Difference Scheme to approximate BVP

I have the following boundary value problem: $\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}=0, \; 1<x<2, \;0<y<1;$ $u(x,0)=2\ln x, \; u(x,1)=\ln(x^2+1), \; 1 \leq ...
1
vote
1answer
92 views

Finite difference method stability

I have shown that a finite difference method satisfies $$\underline{u}^{n+1}=((1+6\mu)\mathbb{I}-36\mu A^{-1})\underline{u}^n$$ I don't think that the rest of the question is necessary but it is ...
3
votes
1answer
232 views

Finite difference implicit schema for wave equation 1d not unconditionally stable?

The wave equation 1D with constant density is defined as: \begin{equation} \frac{\partial^2 U}{\partial t ^2} = V^2 \frac{\partial^2 U}{\partial x ^2} \label{eqa} \end{equation} And the implicit ...
1
vote
1answer
226 views

Von Neumann Stability Analysis

I came across the following task recently: Use the von-Neumann stability analysis to investigate the stability of the discrete form of $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial ...
0
votes
1answer
140 views

finite differences for PDE's

I am having trouble with this question, much appreciated if anyone can help? a) for the 2nd order wave equation: \begin{align} &\partial_{tt}u(x,t)-c^2\partial_{xx}u(x,t)=0 & (x,t) \in ...
2
votes
0answers
66 views

How to can I transform the 2D cuasi Laplace equation with variable coefficients to finite difference scheme?

I want to solve $$\frac{\partial}{\partial x}\left(\frac{1}{\rho(x,y)}\frac{\partial \Phi}{\partial x}\right) + \frac{\partial}{\partial y}\left(\frac{1}{\rho(x,y)}\frac{\partial \Phi}{\partial ...
1
vote
0answers
169 views

Reference for Finite Difference Schemes

Is there any place that I can find a list of different PDEs and common finite difference schemes used for each? I have seen tables of finite difference coefficients such as the one here ...
4
votes
1answer
153 views

Stability of pde in some $L^p$ norm and stability of a numerical scheme for it equivalence.

I would like to get some light on how to proceed and my confusion. I consider some IBVP of the form $$u_t+L(t,x)u=0, x\in D, t\in [0,T]$$ with some BC and initial data. And I use some numerical method ...
1
vote
1answer
227 views

Harmonic functions and conformal mappings

I would like to get some insight into the practicalities of applying conformal mapping techniques for the numerical solution of PDEs. Up until now I had the impression that conformal mapping ...
1
vote
0answers
111 views

Orthogonal vs general curvilinear coordinates

Solutions to PDEs over irregular domains can be computed using the finite difference method by the introduction of so called body fitted coordinate systems where the coordinate lines are aligned to ...
4
votes
1answer
377 views

When do the solutions of a system of difference equations converge to solutions of a system of PDEs?

I have a set of rather nasty, nonlinear difference equations roughly of the following form: $$ \frac{a^{(j)}_{i+1}(s)-a^{(j)}_i(s)}{\epsilon}=f^{(j)}(\lbrace a^{(j)}_i(s)\rbrace_{j=1}^m,\lbrace ...
1
vote
1answer
89 views

Evaluating partial derivatives to apply finite differences

I need to solve the following partial differential equation (Cahn-Hilliard) using finite differences: $$\frac{\partial c}{\partial t} = \nabla^2h + \cdots$$ where $h = c(1-c)(1-2c)$. The question I ...
0
votes
1answer
78 views

How do I solve this pde (using finite differences)?

How do I solve the pde: $\ -s_x(x,t) -p(x,t)s_t(x,t)=p(x,t)$ for s(x,t) when p(x,t)=2x, subject to the condition s(0,t)=0? Generally p(x,t) may not be analytical so I would like to use finite ...
1
vote
1answer
169 views

How do I combine two matrix equations into one?

I have a discretely sampled 2D function: S = 1 2 3 4 1 2 3 4 1 2 3 4 I want to find finite difference ...
2
votes
1answer
284 views

How do I solve an overdetermined linear system of partial differential equations?

I have two partial differential equations that I want to solve (for $\ \sigma $) by finite differences: $\ -\frac{\partial \sigma}{\partial x}(x,y,t) -p(x,y,t)\frac{\partial \sigma}{\partial t}(x,y,t) ...
1
vote
1answer
207 views

norm for estimating the error of the numerical method

In most of the books on numerical methods and finite difference methods the error is measured in discrete $L^2$ norm. I was wondering if people do the in Sobolev norm. I have never see that done and I ...
2
votes
1answer
68 views

transport equation FD methods

Please help me to understand the following: if I have a transport equation $u_t+au_x=0$ and I want to solve it using finite differences I can see a lot of info on the explicit with central differences ...
1
vote
0answers
306 views

Stability condition for explicit scheme in finite differences

I've the following explicit scheme in finite differences (for a one dimensional non uniform diffusion problem), being $k$ the time step, $h$ the space step, $A$ the thermal conductivity at position ...