# Questions tagged [numerics]

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### 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 ...
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 ...
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
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### 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 ...
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 ...
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### 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 (...
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 ...
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: u_f = \overline{u}_f - \overline{D}_f\...
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 ...
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 ...
210 views

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### 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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
2k views

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### 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 ...
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 $n\geq0$ ...
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 ...
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 ...
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 ...
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### 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}...
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### 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 ...