Questions tagged [numerics]

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votes
1answer
82 views

Is iteration an efficient algorithm in this case? [closed]

My task in numerical analysis is We are interested in finding values of β0 for which z(x) = 2500. Use an efficient algorithm to determine the rays which pass through the receiver. Now I'm just ...
2
votes
1answer
130 views

Finding the frequencies of vibration of a circular and square drum

I want to find the frequencies of vibration of a circular and square drum. To do this, I need to solve a 2-dimensional wave equation (PDE) with boundary conditions. Every method that I have researched ...
-1
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1answer
388 views

Finite Difference for Fourth-Order PDE

How to discretize the following 4th order PDE using finite difference method? $$\frac{\partial^{2} y}{\partial t^{2}}+\frac{\partial^{4} y}{\partial x^{4}}=0$$ thanks
6
votes
1answer
756 views

Stability analysis of Heun's method

I am using Heun's method with a third order upwind spatial scheme, which is suggested by Shao (2008) to be used for solving the horizontal advection part of the advection-diffusion equation. This is ...
8
votes
2answers
128 views

How can I numerically solve an ODE to $N$ provably correct digits?

Suppose we have an initial value problem of the form $$ \frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t} = f(\mathbf{x}) \qquad \mathbf{x}(0) = \mathbf{x}_0 $$ where $\mathbf{x}_0 \in \mathbb{R}^n$ is known ...
1
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0answers
131 views

Numerical integration when solving PDE: Simpsons rule and high frequency noise [closed]

I am solving a PDE, and one of the intermediate steps is to numerically integrate a function over a compact interval. The function is represented on a linearly spaced grid. I am using Simpsons rule (...
2
votes
1answer
449 views

Hessian-free and Truncated Newton methods

In this paper on Deep Learning for Machine Learning, the approach is referred to as Hessian-free method. That is because the Hessian is never computed explicitly. Instead, the product of the Hessian ...
5
votes
2answers
748 views

A Question About the Rhie-Chow Interpolation Used for Solving the Incompressible Navier-Stokes Equations on Unstructured Grids

When using the SIMPLE method on a mesh with a collocated variable arrangement, the following interpolation is used for the advecting velocities: \begin{equation} u_f = \overline{u}_f - \overline{D}_f\...
3
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1answer
79 views

Resource recommendations for numerical methods involved in dynamical systems analysis

I am interested in learning numerical methods that specifically have to do with analyzing dynamical systems. In particular: drawing phase plane diagrams drawing phase portraits analyzing bifurcations ...
1
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0answers
167 views

Tips on improving stability in numerical scheme for non-linear PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
8
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3answers
210 views

Stability criterion for waves in anisotropic solids

The equations of motion for an elastic solid are given by $$\begin{align} &\nabla \cdot \boldsymbol{\sigma} + \mathbf{f} = \rho \ddot{\mathbf{u}}\\ &\boldsymbol{\sigma} = \mathbb{C}\...
0
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1answer
99 views

Spectral Coefficients of Implicit Finite Difference Solution

This is something I've been trying to figure out for a long time, and all I have is vague numerical results. I'm trying to answer the following question analytically: Suppose I have a time dependent ...
0
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2answers
98 views

Explain this multivariate differential identity

$$ \frac{\partial|\nabla\phi|^2}{\partial\phi}=-2\nabla\cdot\nabla\phi$$ I would very appreciate that you help me . Please do it in detail, I am quite not good at such problems. There is something ...
3
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0answers
373 views

Calculate Integral Using Gauss Jacobi Quadrature or otherwise

I need to integrate the following integral: \begin{align} I = \int^z\frac{1-\zeta^2}{(1+\zeta^2)(\zeta-\zeta_l)(1-\zeta_l\zeta)}\prod_{k=2}^{n-1}\left ( \frac{\zeta-z_k}{1-\zeta z_k} \right )^{-\...
3
votes
3answers
220 views

Integration over a complicated domain

Consider points $(x,y)$ in a domain $\mathcal{D}$ that is bounded above by a curve $C$ that is multivalued (i.e. I have points defining it that are of the form $(x_n,y_n)$, $n=1,2,..,N$), and looks ...
7
votes
1answer
222 views

Compute eigenvectors of a matrix with known eigenvalue spectrum

If I have already accurately known the eigenvalue spectrum (i.e. all eigenvalues) of a matrix, is there any efficient numerical algorithm to compute all the eigenvectors corresponding to these ...
1
vote
1answer
115 views

Accurate way for computing a ratio coming from Monte Carlo simulation

I am seeking recommendations on how to compute the Binder ratio numerically accurate when doing Monte Carlo simulation on spin models. Binder ratio is defined as: $$ B = \frac{\langle M^4\rangle}{\...
3
votes
1answer
170 views

Rounding errors in images of Julia sets

One typically computes Julia sets by iterating a complex function, such as a polynomial or rational function. How do rounding errors affect the results? I'm looking for references on this issue, ...
6
votes
1answer
349 views

Caveats of Hessian free method

Hessian free iterative optimization techniques like Newton-CG, do not explicitly compute the Hessian but instead approximate the product of the Hessian with a vector through finite difference. The ...
8
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2answers
1k views

Does the matrix condition number affect accuracy of iterative linear solvers?

I have a rather specific question regarding the condition number. I run FEM simulations which have multiple length scales to them which results in a huge disparity between the largest entries and the ...
6
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4answers
840 views

Levenberg-Marquardt - What is preferable (A + mu.I) or (A + mu.diag[A])?

The step size is computed by solving $$ (A + \mu I) h = -g $$ I could find in some literature that one can compute the step size by solving $$ (A + \mu \operatorname{diag}(A) ) h = -g $$ It is said ...
1
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0answers
176 views

Am I using the incorrect implementation of the fast Chebyshev transform?

I was told that the fast Chebyshev transform has superior spectral convergence, but I am unable to verify its rumored convergence. I was given plots of its spectral convergence, where the signal's ...
11
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1answer
7k views

Applying the Runge-Kutta method to second order ODEs

How can I replace the Euler method by Runge-Kutta 4th order to determine the free fall motion in not constant gravitional magnitude (eg. free fall from 10 000 km above ground)? So far I wrote simple ...
2
votes
1answer
800 views

Doubt regarding stopping criterion for Newton method

I am solving an unconstrained convex optimization problem, which can easily have a million variables. I am trying to get a working system with a toy problem of around 200 variables. I am noticing that ...
2
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0answers
539 views

What is the difference between SSPRK3 and RK3 time discretization methods?

Here, SSPRK3 refers to third order strong stability preserving Runge-Kutta and RK3 refres to regular third order Runge-Kutta method. The meaning of the method is obvious from the name. However there ...
0
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1answer
202 views

Strict Feasibility in Interior Point Methods

As we know, in the interior point methods, all the iterates have to be strictly feasible. I implemented an affine scaling interior point for nonlinear objective functions. For small examples (2D), it ...
0
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2answers
2k views

How to obtain a convergent solution iteratively for a linear system of equations? [closed]

I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is: $$\underbrace{\begin{bmatrix} r_{11} & r_{12} &...
6
votes
1answer
349 views

roots of polynomials with small coefficients

I would like to compute the roots of a polynomial with exponentially small coefficients. $$ \sum_{n=0}^N a_n \frac{z^n}{\sqrt{n!}} \tag{$\ast$}$$ where $a_n$ are Normal random variables with mean $0$...
1
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3answers
168 views

Numerical integration algorithm for this set of ODEs

I have to solve numerically the following initial value problem, which physical origin is discussed in this article by G.I. Taylor (1941). Here $\eta$ is the independent variable and ranges beetween $...
8
votes
1answer
2k views

Newton iteration for cube root without division

It's a fairly well known trick to avoid division in calculating square-roots to apply Newton's method to finding $1/\sqrt{x}$, and probably better known, using Newton's method to find reciprocals ...
1
vote
1answer
188 views

What does “strongly conservative” mean in the context of numerical methods?

I have a homework problem that asks me to show that 1st order unwinding or central differencing can give a strongly conservative, consistent scheme for the 1-D Burger's Equation using a finite volume ...
2
votes
1answer
466 views

Laplacian discretization for parametric curves

I know how to compute the discrete Laplacian of a graph and of a mesh (the Laplace-Beltrami operator). Is there an analogous definition for the computation of the Laplacian of a parametric curve ? ...
5
votes
2answers
741 views

How to impose boundary conditions on eigenfunction problems?

I am trying to solve for the eigenfunctions of a (1D) differential operator using finite differences: $$A \, f(x) = \lambda f(x)$$ Here is an example in Python where $A = \partial_x^4$: ...
2
votes
1answer
439 views

Derivation of Adams-Bashforth coefficients

From the order condition $$\sigma(w)=\frac{\rho(w)}{\ln w}+O(|w-1|^p)$$ I get $$\sigma(w)=1+\frac{3}{2}(w-1)+\frac{5}{12}(w-1)^2=-\frac{1}{12}+\frac{2}{3}w+\frac{5}{12}w^2$$ Those coefficients are ...
8
votes
1answer
313 views

Good tutorials on how to use Butcher tables?

I tried to go to the primary sources in order to understand how to use Butcher tables to simplify the algebra I need to do when using Taylor series to find the order of accuracy of a scheme, for ...
4
votes
1answer
233 views

Methods for Constrained Optimization Problems with Box Constraints

Consider this problem: \begin{equation} \begin{array}{ll} \text{minimize } & f(x) \\ \text{subject to } & a \leq x \leq b \end{array} \end{equation} where $a,b,x \in \mathbb{...
3
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1answer
291 views

Active Elements in Projected Newton's Method?

To those who are familiar with the projected Newton's method or projected gradient method... We consider a constrained optimization problem with simple bounds. Particularly, minimize f(x) subject to ...
0
votes
1answer
73 views

How to integrate Euler Bending Equation in C++? [closed]

I am trying to Draw shear force diagram and bending moment diagram of beams. In this, I need to integrate second order differential. So, Anybody can suggest me, which numerical method should I use?
4
votes
2answers
474 views

Gradient of a vector on a triangulated lattice

Suppose I have a triangulated mesh and I have solved a differential equation problem and have a solution phi. From phi I can construct a vector field, how would I go about computing the gradient of ...
5
votes
1answer
121 views

What is lost in terms of approximation, when writing a problem in terms of a Differential Algebraic Equation (DAE) system rather than an ODE system?

It may be that we have a model where the following equation holds for some phenomenon: $$(1)\quad x + y + z = T$$ Importantly, $T$ is a constant, i.e.: $$(2) \quad \frac{\mathrm{d}T(t)}{\mathrm{d}t} ...
2
votes
3answers
202 views

Compute accuracy order as mesh gets refined?

I have implemented a FVM code and now I need to plot the accuracy of the method as the mesh gets refined. Having a very fine mesh, my idea is to compare what is the error between the coarser and fine ...
0
votes
1answer
440 views

Deriving the error bound for Bisection Method

This is a homework question, I would like to know if someone can shed some light on it. Let $x_n = \frac{a_n + b_n}{2} , r=\lim_{n \to \infty}x_n$ and $e_n =r-x_n$. Here $[a_n,b_n]$ with $n\geq0$ ...
0
votes
1answer
107 views

Computing multiple numerical derivatives at once

Lets say I have a function $f(X) = f(x_1,...,x_N)$ to be integrated. But unlike time discrete methods, my integrator uses quantisation to advance time, that is if $|x - q| > dQ$, with $q$ being the ...
-1
votes
1answer
114 views

Can you give some information for rothe method [closed]

I want to learn a numerical method for PDEs other than finite difference method. After some research on internet i have found Rothe method and it looks interesting to me. Unfortunately, i couldn't ...
0
votes
1answer
258 views

Numerical Solution of non-linear diffusion equation using Finite Differencing

I'm trying to solve the following non-linear diffusion equation: $$ \frac{\partial}{\partial t} u(x,t)= \frac{\partial^{2}}{\partial x^{2}}u(x,t)^{3}$$ $$ -1\leq x \leq1, t \geq 0 $$ with the boundary ...
2
votes
1answer
189 views

Unable to validate the Roe matrix for the Shallow Water Equations

In LeVeque's Finite Volume Methods for Hyperbolic Problems, p. 320-321, one may find the derivation of the Roe matrix to the 1D Shallow Water Equations (SWEs). It is $$ \hat{A}_{i-1/2}=\begin{pmatrix}...
3
votes
1answer
140 views

Computing expectations

I want to compute the following conditional expectation $E_{t}[\phi(A_{t+1}, \eta_{t+1})| A_t]$ where $\log A_{t}=\rho \log A_{t-1} + e_{t}$ and $e_{t}$ is IID $N~(0,\sigma_e)$ and $\eta_{t}$ is ...
0
votes
1answer
164 views

Neumann boundary problem

I'm writing a solver for a differential equation with two neumann boundaries (u'(0)=u'(1)=0) and I can't figure out how to determine how to solve the problem. What will my boundaries be and how do I ...
3
votes
0answers
107 views

How to tell if symplectic integrator is more suitable for my problem, and what are downsides?

This question follows another one that I have already asked. My intention was to use a classical Runge-Kutta 45 method to solve ODEs of my system. However, I have seen recommendations for using ...