0
votes
0answers
22 views

Checking if the Hessian is the derivative of the gradient

Suppose f: R^n --> R. I have a code that computes the gradient of f. I have another code that computes the Hessian of f times a vector. Now I want to check if they are correct. Specifically, I am ...
1
vote
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 ...
2
votes
0answers
97 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 ...
0
votes
2answers
39 views

Avoiding substraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the substraction is blown up for small h. This I can verify ...
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} + ...
2
votes
0answers
81 views

Boundary integral method to solve Poisson equation

Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential ...
1
vote
1answer
27 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
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. ...
3
votes
1answer
94 views

Having trouble understanding the finite element method

I am trying to read some sources on how to implement the finite element method and I am having difficulty putting all the concepts together. I can read and understand the Galerkin approach just fine. ...
0
votes
0answers
38 views

Finite difference discretization of the Porous Medium Equation - How to prove stability?

The porous medium equation is given by $\delta_t u(x,t) = (u^p \cdot u_x)_x, (x,t) \in \mathbb{R} \times (0, \infty)$ We are working on approximating the solution with finite differences. ...
4
votes
1answer
26 views

Numerical computation of the $n^{\mathrm {th}}$ derivative of a multivariate function

From a multivariate function $f$, depending on $n\geq 1$ variables, which can be computed numerically, but which does not admit simple analytic expression, I would like to approximate numerically the ...
1
vote
0answers
68 views

Calculate a 5x5 Vandermonde system for a 5 point mesh

This is problem 1.2 in Randall J Leveque's textbook, "Finite Di fference Methods for Ordinary and Partial Di fferential Equations". I'm struggling with how to actually do the computation, I'm not so ...
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 ...
2
votes
1answer
104 views

Finite element method for the 'Particle-In-a-Box' problem in quantum mechanics

(Apologies in advance for the lengthy question, but it really is needed for a precise description of what I've done!) In suitable units, the 'Particle-in-a-box' problem is described by the following ...
2
votes
2answers
174 views

Derive forward Euler method for two-variable function

I need to derive the forward Euler method for solving ODEs and I would like some comments on what I have so far; overdot denote the time derivative: $\dot x \overset{def}{=} dx/dt$. Say we have ...
0
votes
0answers
197 views

Solve the following boundary value problem using the finite difference method.

Solve $$y''=\frac{1}{2}y'-\frac{1}{2}y+\frac{x^2+3}{2}, ~~~~~y(0)=1, ~~y(4)=24$$ using the second order finite difference approximation order with $h=1$. I know that we use ...
1
vote
1answer
37 views

Chain rule in finite calculus for ODEs (RK methods)

SE.Math, I'm reviewing for a test on numerical methods and I am having some difficulty in understanding how to take the second derivative in finite terms. I have: $$ U(t_k) $$ as my underlying ...
0
votes
0answers
60 views

A 2D secant method?

I've recently had occasion (providing an engineering colleague with a little mathematical help) to solve a non linear system $\begin{align*}f(x,y)&=0,\\ g(x,y)&=0.\end{align*}$ If ...
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
1answer
210 views

Applying two-point forward to two-point forward formula

What do you get when you apply the two-point forward finite difference formula for the first derivative of $f(x)$ to the two-point forward finite difference formula for the first derivative of ...
0
votes
0answers
530 views

Finite difference approximation of heat equation with source term

I am using the implicit finite difference method to discretize the 1-D transient heat diffusion equation for solid spherical and cylindrical shapes. The general equation is: $$ ...
2
votes
2answers
162 views

Is there a mean value theorem for higher order differences?

The standard mean value theorem tells us $\frac{f(x+h)-f(x)}{h} = f'(c)$ for some $c$ between $x$ and $x+h$. Rewriting this, we may see it as $\frac 1h\Delta_h f(x) = f'(c)$. This makes me wonder if ...
1
vote
1answer
55 views

Numerical solution of a difference equation

I would like to solve in $f$ the equation $f(x) - f(x-d) = g(x),$ where $g$ is a given function and $d$ a given constant delay. We can assume $f(x) = 0$ for negative $x$. When $g$ is sampled (with a ...
1
vote
1answer
187 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: ...
2
votes
1answer
131 views

Related to Applying Runge-Kutta Method

I have an initial value problem (henceforth IVP) as follows: $$\frac{d \Phi(t)}{dt}= A(t)\Phi(t)$$ subject to the initial condition $\Phi(t_0)=I$, where $\Phi(t), A(t), I$ are square matrices of same ...
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 ...
1
vote
1answer
122 views

Finite Difference without boundary conditions

I'm working through the paper where the Finite Difference method is employed to solve the PDE $\displaystyle \frac{\partial u(x,t)}{\partial t} = a \cdot \frac{\partial^2 u(x,t)}{\partial x^2} + b ...
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
509 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 ...
2
votes
0answers
101 views

One Sided Approximation for Mixed Derivatives

Consider the function u(x,y,z) I am trying to approximate the partial derivative at point (i,j,k) by one sided finite difference method. Now using one sided 2nd order finite difference approxmation ...
1
vote
1answer
603 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: ...
2
votes
1answer
62 views

Interpolation with degree restriction

(using $f[x_1, ... , x_n]$ to denote the forward difference operator) I have a polynomial $P(x)$ interpolating $5$ points $x_0, ... , x_4$ and $2$ derivative values $x_0, x_3$ across an evenly spaced ...
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
227 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 ...
0
votes
1answer
427 views

linear shooting method and finite differences

how can we use the linear shooting method to approximate this solution below: $$y'' + 4y = \cos(x), 0 \le x \le4, y(0) = 0, y(pi/4) = 0, h = \frac{\pi}{20}$$ My concern is with the RK-4 and setting ...
2
votes
0answers
68 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
239 views

Solving ODE numerically with central difference quotient

I try to understand an old Fortran code that is not well documented. In this code the ODE $$ \frac{dy}{dx} = -\frac{B(x - y)}{y} $$ is solved numerically as an initial value problem from $x_0=0.99$, ...
3
votes
2answers
197 views

$\Delta^ny = n!$ , difference operator question.

I was looking in a numerical analysis book and found the statement: if $y=x^n$ and the difference is $h=1$ then $\Delta^ny = n!$ and $\Delta^{n+1}y = 0$. Here $\Delta y = y(x+1)-y(x)=(x+1)^n -x^n$, ...
1
vote
0answers
170 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 ...
1
vote
1answer
260 views

Error estimator for forward finite difference

I'm studying numerical analysis and for the approximation of a derivative around a given point we have for the forward finite difference: $$(\delta_+f)(\bar x) = \frac{f(\bar x + h) - f(\bar x)}{h}$$ ...
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
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 ...
1
vote
1answer
62 views

Determine the numerical method

Please, help to understand the method which is used in the following snippet: ...
2
votes
0answers
224 views

linear interpolation error estimate for non-smooth function

Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be ...
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
287 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 ...