# Questions tagged [numerics]

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### Which Runge-Kutta method is more accurate: Dormand-Prince or Cash-Karp?

I simply want to know whether the Dormand-Prince Numerical Method or the Cash-Karp Numerical Method is more accurate.
14k views

### How should boundary conditions be applied when using finite-volume method?

Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh, I would like to apply a Robin type boundary condition to the l.h.s. of the domain (...
4k views

### What's the state-of-the-art in highly oscillatory integral computation?

What's the state-of-the-art in the approximation of highly oscillatory integrals in both one dimension and higher dimensions to arbitrary precision?
631 views

### How to check the correctness of my implementation of a numerical scheme for differential equation

I know the method of constructing solutions. For example I have a BVP: $$u_{xx} + u = 0$$ subjected to: $$u_{x}(0) = f_1,$$ $$u_{x}(1) = f_2$$ If I want to check the correctness of my ...
716 views

### Finite difference discretization on a circle

I am trying to discretize the differential operator $\frac{d^2}{dx^2}$ acting on $S^1 = [0,1]$ using finitely many points around a circle at $0, \frac{1}{N}, \frac{2}{N}, \dots, \frac{N-1}{N}$. Here ...
138 views

### Calculation of the EFIE integral

I need help computing the following integral: $$\int_{}\frac{(1+jk|\vec{r}-\vec{r}^\prime|)e^{-jk|\vec{r}-\vec{r}^\prime|}}{|\vec{r}-\vec{r}^\prime|}d\vec{r}^\prime$$ in this integral $\vec{r}$ ...
459 views

### Interpolating a mathematical function using a Hermite Cubic Finite Element Space

I have a Hermite Cubic Finite Element Space on a computer in the form of Matlab m-files. More specifically, I can evaluate four "shape functions" $N_1, N_2, N_3,$ and $N_4$, for which the following ...
172 views

1k views

### Scientific standards for numerical errors

In my field of research the specification of experimental errors is commonly accepted and publications which fail to provide them are highly criticized. At the same time I often find that results of ...
4k views

### What's the state of the art in parallel ODE methods?

I'm currently looking into parallel methods for ODE integration. There is a lot of new and old literature out there describing a wide range of approaches, but I haven't found any recent surveys or ...
1k views

### Conserving Energy in Physics Simulation with imperfect Numerical Solver

I am creating a C++ Physics Simulation where I need to move an rigid body through an acting force field. Problem: simulation does not conserve energy. Quesiton: abstractly, how is conservation of ...
8k 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 ...
397 views

### Oscillations in singularly perturbed reaction-diffusion problems with finite elements

When FEM-discretizing and solving a reaction-diffusion problem, e.g., $$- \varepsilon \Delta u + u = 1 \text{ on } \Omega\\ u = 0 \text{ on } \partial\Omega$$ with $0 < \varepsilon \ll 1$ (...
204 views

### Which optimization method can be used to do the following?

I've the following system of equations for studying information flow in the below graph, $$\frac{d \phi}{dt} = -M^TDM\phi + \text{noise effects} \hspace{1cm} (1)$$ Here, M is the incidence ...
1k views

### How do you debug numerical code, what could be source of this oscillatory error?

Quiet a lot of insight can be gained form experience, I was just wondering if anybody has seen something similar to this before. The plot shows the initial condition (green) for the advection-...
8k views

### Efficient computation of the matrix square root inverse

A common problem in statistics is computing the square root inverse of a symmetric positive definite matrix. What would be the most efficient way of computing this? I came across some literature (...
5k views

### Integral in log-log space

I'm working with functions which, in general, are much smoother and better behaved in log-log space --- so that's where I perform interpolation/extrapolation, etc, and that works very well. Is there ...
163 views

I'm interested in studying the effect of mesh size on the behavior of the solution curves of 1D convection-diffusion problem. $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - ... 1answer 2k views ### How do I form the Chebyshev differentiation matrix in MATLAB? I have some code that does exactly this, but I do not like to use things I do not understand. Here is the code ... 2answers 2k views ### How to plot orbit of binary star and calculate its orbital elements? I have a set of dates, position angles (\theta) and angular separations (\rho) for visual binary star. For example: ... 4answers 705 views ### Relevance of fixed-point and arbitrary precision computations I see very few non-floating point computing libraries/packages around. Given the various inaccuracies of floating point representation, the question arises why there aren't at least some fields where ... 2answers 1k views ### What does the Von Neumann's stability analysis tell us about non-linear finite difference equations? I am reading a paper [1] where they solve the following non-linear equation $$u_t + u_x + uu_x - u_{xxt} = 0$$ using finite difference methods. They also analyse the ... 1answer 142 views ### Numerically estimating expected value of f(x) when x is normally distributed I need to estimate$$ \mathbb{E}_x[f_i(x)] = \int_{\mathbb{R}^n} f_i(x) p(x) dx $$for many functions f_i(x), where p(x) is the density of a normal distribution. The evaluation of all the ... 0answers 333 views ### Frozen coefficient method (von Neumann stability analysis) Earlier it was considered that frozen coefficients method for Neumann stability analysis for finite difference scheme is more heuristic than rigorous. But I have read some information in a book by ... 1answer 801 views ### ground state from the Schroedinger equation with a central potential what happens to the origin I have code that attempts to implement a solution to the Schrödinger equation where there is a central potential (more or less im thinking of hydrogen), in 1-D using the numerov method to construct ... 1answer 354 views ### Question on how MATLAB's pdepe solver works I'm solving the following 1D transport equation in MATLAB's pdepe solver.$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$At the inlet (left ... 1answer 1k views ### How can I solve wave equation for circular membrane in polar coordinates? The original equation is$$\frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \...
I have difficulties with this equation $$\frac{d^2 u}{d x^2} + u^2 - x^2 = 0$$ with boundary conditions: $u(0)=u(1)=0$ I do not know how to solve nonlinear differential equations with Newton's ...