Questions tagged [numerics]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
1answer
3k views

Newton's method in interpolation

I've seen that in Newton's method for interpolating polynomials, the coefficients can be found algorithmically using (in Python-ish): ...
15
votes
1answer
11k 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 (...
1
vote
3answers
1k views

Capacitance in freefem++

I would like to simulate a capacitor in 2d with freefem++. This is the code I used: ...
2
votes
2answers
243 views

How can i solve this first order system of differential equations

Edit: I am trying to solve $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu\frac{\partial^{2}u}{\partial x^{2}}\\ u(0,t)=0\nonumber \\ u(1,t)=0\nonumber \\ u(x,0)=\sin(\pi x)\\ 0<...
3
votes
1answer
115 views

Efficient computation of tangent of fraction of angle

I want to compute $a = \tan(f \theta)$ for $f\in [0,1]$, given $g = \tan\theta$. Obviously, I can compute $a = \tan(f\tan^{-1}g)$, but I'm wondering if there's a more efficient way that avoids having ...
4
votes
0answers
228 views

Negative viscosity stabilized by fourth order terms

I am trying to solve a "Navier-Stokes"-type problem where the viscosity is negative. Of course this renders the equation unstable and thus I add a fourth order term, so the entire equation becomes: $$...
5
votes
1answer
171 views

Local truncation error and transformation of coordinates

I am given the advection equation $$ u_t=u_x $$ and then the transformation of coordinates $$ x=x(\xi,\theta), \qquad t=\theta $$ which leads us to the transformed equation $$ x_{\xi} u_{\theta} - u_{\...
1
vote
0answers
159 views

Numerically evaluate 1D inhomogeneous wave equation solution

I am trying to solve the following 1D inhomogeneous wave equation. Forgive me if I a miss any rigorous mathematical concept. $$ \frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 u}{\...
4
votes
1answer
206 views

How can exponential fitting be used with the finite element method?

Restricted to one dimensional problem, is it possible to dynamically adapt the finite element method (FEM) discretisation based on the local value of the Péclet number ($P_e$) for advection-diffusion ...
5
votes
2answers
2k views

Does the finite element method impose any restrictions on the Peclet number for numerical stability?

Background on finite volume method When discretising the flux with a central difference stencil of the the advection-diffusion equation restriction $\frac{ah}{d} < 2$ must be observed for the ...
3
votes
1answer
88 views

numerical inaccuracy ellipsoid-ellipsoid collision

I am trying to implement ellipsoid-ellipsoid collision in my C++ code. Briefly this task can described as next: Let's assume that we have two arbitrarily oriented ellipses in the in space and this ...
2
votes
1answer
1k views

Implicit heat diffusion with kinetic reactions

I am using the implicit finite difference method to discretize the 1-D transient heat diffusion equation for solid spherical and cylindrical shapes: $$ \frac{1}{\alpha}\frac{\partial T}{\partial t} = ...
0
votes
1answer
140 views

Simulating the motion of a elastic body under gravity [closed]

I am doing a numerical simulation of a elasticity problem. It is very simple. A cuboid elastic body with the right end fixed on the wall, under the gravity(but here I set it to be 1 along the z-axis ...
5
votes
1answer
388 views

When is discrete Fourier transform a good approximation to the continuous one?

I am looking for help in understanding the use of discrete Fourier transform as an approximation to continuous Fourier transforms. As an exercise, I considered a Gaussian $$f(x) = \exp\left(-2x^2\...
4
votes
1answer
2k views

Analytic solution 2D scalar wave equation in cylindrical coordinates numerical implementation

I am trying to compare my finite difference's solution of the scalar (or simple acoustic) wave equation with an analytic solution. For that purpose I am using the following analytic solution ...
6
votes
2answers
197 views

Continuation procedure to solve for a 2D curve that satisfies f(x,y) = 0

I have some function of $R^2$, that must be numerically computed. For instance, I might be interested in a real-valued contour integral that begins from (x,y) = 0. $$ f(x,y) = \Re\left[\int_0^{x + iy}...
6
votes
3answers
974 views

Significance of p-convergence studies

Consider a method (e.g., FEM) with variable approximation order $p$. Now, we know that the optimal order of convergence is given by $$e = C h^{p+1},$$ where $h$ denotes the mesh size and a constant $C$...
15
votes
2answers
2k views

Is it possible to solve nonlinear PDEs without using Newton-Raphson iteration?

I am trying to understand some results and would appreciate some general comments on tackling nonlinear problems. Fisher's equation (a nonlinear reaction-diffusion PDE), $$ u_t = du_{xx} + \beta u ...
2
votes
0answers
121 views

Lapack++ for QR algorithm

I have recently started using Lapack++ which I found convenient for my programming purpose, in general. Now, I need to solve a matrix using QR algorithm. I've searched the user manual and I found a ...
5
votes
1answer
175 views

Stabilization of solution to one-dimensional system of PDE

I am trying to solve numerically next PDE system: $$\frac{\partial c}{\partial t}=\epsilon\frac{\partial}{\partial x}(\frac{\partial c}{\partial x}+\rho\frac{\partial \varphi}{\partial x}+\frac{vc}{1-...
14
votes
4answers
1k views

Optimal ODE method for fixed number of RHS evaluations

In practice, the runtime of numerically solving an IVP $$ \dot{x}(t) = f(t, x(t)) \quad \text{ for } t \in [t_0, t_1] $$ $$ x(t_0) = x_0 $$ is often dominated by the duration of evaluating the right-...
13
votes
2answers
1k views

Alternatives to von neumann stability analysis for finite difference methods

I'm working on solving the coupled one-dimensional poroelasticity equations (biot's model), given as: $$-(\lambda+ 2\mu) \frac{\partial^2 u}{\partial x^2} + \frac{\partial p}{\partial x} = 0$$ $$\...
2
votes
1answer
73 views

Can you give me a detailed description of (spetral) deferred correction method?

I have just read "Accelerating the convergence of spectral deferred correction methods". The link is here: http://www.unc.edu/~junjia/papers/sdcgmres.pdf‎. But I wonder how to understand deferred ...
3
votes
1answer
281 views

Finite volume with cell averages vs cell totals for conservation equations

What implementation details need to change if I use a cell average approach rather than a cell total approach for the finite-volume method? For example, consider the conservation law, $$ u_t + \...
23
votes
3answers
3k 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?
4
votes
1answer
230 views

Multivariate numerical integration with a non-uniform grid

I want to approximate the integral: $$ I = \int f(\boldsymbol{x})d\boldsymbol{x} $$ where $\boldsymbol{x}$ is $d$-dimensional. I have a set of non-equally spaced points $\boldsymbol{x}_1, \dots, \...
5
votes
1answer
1k views

Integration of an indefinite integral: matlab precision problem

The integral I need to evaluate is: $$ \int_x^{\infty} \frac{t^n}{e^{t} -1} dt $$ After some research I found a paper saying, The numerical values of the two integrals [...] are easily calculated ...
1
vote
0answers
230 views

Block Backward Differentiation Formula (BBDF), on order 4 formula

I am trying to implement a program the numerical method to solve ODE called Block BDF as explained in this article: https://waset.org/journals/waset/v38/v38-49.pdf As it is variable step-size, I need ...
6
votes
1answer
419 views

ENO-WENO Schemes: Are ENO-WENO schemes non oscillatory for all kinds (linear/non linear) of problems?

Is there an rigorous proof of ENO-WENO schemes being non oscillatory?
12
votes
2answers
346 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 $0 < \varepsilon \ll 1$ (...
3
votes
1answer
200 views

Is using Monte Carlo method a good approach for solving Boltzmann equation?

I'm trying to solve for electron and hole distribution function using Boltzmann equation with various scattering mechanisms. Since I land up with an integro-differential equation, analytical solution ...
11
votes
3answers
1k views

How should non-constant coefficients be treated with finite-volume first order upwind scheme?

Starting with the advection equation in conservation form. $$ u_t = (a(x)u)_x $$ where $a(x)$ is a velocity which depend on space, and $u$ is a concentration of a species which is conserved. ...
4
votes
1answer
547 views

Solving Coupled ODE eigenvalue problem

I've been trying to find some resources that would help me figure out how to numerically solve a coupled system of ODEs which is also an eigenvalue problem. The system is something like: $ \tag{1} ...
3
votes
1answer
273 views

Closed form for singular values of 2D Laplacian?

Does anyone know where to find an analytic form for the singular values of the finite-difference approximation to the 2D Laplacian, expressed in matrix form for a square grid? This would be for the ...
8
votes
2answers
263 views

Astoundingly large difference when evaulating trigonometric identity with NumPy

According to Wolfram Alpha and the Sage computer algebra system, the following identity holds: $$ \cos\left(\arctan\left(\frac{l_1-l_2}{d}\right)\right) = \frac{1}{\sqrt{1 + \frac{(l_1-l_2)^2}{d^2}}} $...
10
votes
4answers
617 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 ...
40
votes
4answers
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 ...
5
votes
2answers
194 views

Imposing invertibility on a Matrix

I have a symmetric positive semidefinite covariance matrix $A$, which is approximately computed as the output of a quadratic regression. I then need to invert $A$, but often it is close to singular. I'...
10
votes
3answers
486 views

Regression testing of chaotic numerical models

When we have a numerical model that represents a real physical system, and that exhibits chaos (e.g. fluid dynamics models, climate models), how can we know that the model is performing as it should? ...
4
votes
1answer
99 views

Bounded Variation Spaces

Could someone explain me (roughly) the interest of Bounded Variation (BV) Spaces for PDEs ? Is there any numerical application of those space to real problems or is it just a theoretic way to ...