Questions tagged [numerics]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
2answers
77 views

Stability of Crank-Nicolson for $u_t = iu_{xx}+2iu$

I want to use the Crank-Nicolson scheme to solve the equation $$u_t = iu_{xx}+2iu$$ Here's the analysis: Suppose we make a grid, with $k = dt$ and $h = dx$, the usual notation, and also $u_j^n = u(...
0
votes
1answer
63 views

Fast Poisson solver (with Dirichlet BC zero) on a *truncated* Cartesian 3D grid

I find myself in the position of having to solve $-\Delta u = f$ on a subset of Cartesian grid points that don't necessarily form a cuboid domain subject to a homogenious Dirichlet boundary condition ...
0
votes
0answers
58 views

The final Boundary Condition is Unknown, Is Backward Euler is still valid to be implemented?

I am working on conductive polymer modeling and supposed to do one-dimensional diffusion model in the thickness of the polymer, however, due to the small value of thickness in micro, when I use the ...
5
votes
0answers
82 views

Numerical methods for the continuity equation with Sobolev vector field

Consider the continuity equation $$ \partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N, $$ with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
1
vote
0answers
33 views

Why does the correlation function of this stochastic differential equation starts at different points?

I am working with the following differential equation: The equation is $$x=\beta +\sqrt{2D} \xi(t)$$ where $\xi(t)$ is a white noise term, with a reflecting wall boundary conditions. After solving ...
2
votes
0answers
72 views

How to implement adaptive step size Runge-Kutta Cash-Karp?

Trying to implement an adaptive step size Runge-Kutta Cash-Karp but failing with this error: ...
0
votes
0answers
40 views

Time sampling changes solution

I'm currently trying to solve a problem using numerical methods. The set-up is rather long, so I apologize in advance... TL;DR: My solutions change depending on how big my steps are and I don't know ...
3
votes
4answers
266 views

Numerical integration in Python with unknown constant

I’d like to solve the below equation for the unknown $T$: $$\int_0^\infty \frac{x^2}{\exp\left(\frac{x}{T}\right)-1}\kappa_x \mathrm{d}x = C,$$ where $C$ is a known constant and $\kappa_x$ is some ...
0
votes
0answers
63 views

Need help applying Implicit Eulers Method together with Newtons Method on Burgers' Equation

From the inviscid Burgers' equation: $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x} = 0$, I get the discretization $\frac{u_i^{n+1}-u_i^n}{\Delta t}+\frac{(u_i^{n+1})^2-(u_{i-1}^n)^2}{\...
2
votes
1answer
115 views

Why the numerical solution of advection-dominant problem is challenging

In many CFD text books, usually there is a dedicated chapter for advection term discretization. Why discretization of such term in advection-dominated problems and near the discontinuities is ...
1
vote
1answer
61 views

Is there a better way to do run time analysis than this?

I currently have 2 different functions with options to vectorise them: acc_rej_sine(max_iter, algorithm=None) and ...
9
votes
5answers
3k 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 ...
1
vote
1answer
84 views

Approximation Error in a Finite Difference Approximation of the Square of Derivative

First Part: (First-order derivative) Assuming $f$ is an infinitely differential function everywhere, the Taylor series of $f(x + h)$ at $x$ is \begin{align}\tag{1} f(x + h) = f(x) + hf'(x) + \frac{1}...
5
votes
1answer
3k views

What is numerical damping in the context of time-dependent FEM solvers?

Comsol Multiphysics (a popular FEM package) includes two time-stepping algorithms (IDA aka BDF, and Generalized-alpha), described in their documentation as follows (quoted here under Fair Use; ...
2
votes
1answer
168 views

Mass Matrix and how to handle it (ODEs) - References

I'm interested in a good reference/paper about how to handle numerically a mass-matrix system as \begin{align} \mathbf{M}(t,y)\dot{y} =F(y,t) \end{align} I know that such a problem can be solved by ...
0
votes
0answers
69 views

Analytic vs discrete understanding of PDE

The PDE I am working with: $$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\ \partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\ \beta(x)>0$$ Integrate the PDE: $$\int_\Omega \partial_t u=\...
3
votes
1answer
135 views

What does the exponential function mean in numerical ODE solving formulas?

I'm trying to read papers on numerical ODE algorithms and I always seem to stumble upon huge amounts of exponentials multiplied by each other. For example in New families of symplectic splitting ...
10
votes
1answer
140 views

Are there shortcuts for numerically approximating systems of ordinary differential equations when autonomous?

Existing algorithms for solving ODEs handle functions $\frac{dy}{dt} = f(y, t)$, where $y \in \mathbb R^n$. But in many physical systems, the differential equation is autonomous, so $\frac{dy}{dt} = f(...
0
votes
1answer
84 views

Actual global error vs theoretical global error: How to combine theory with practice

I have implemented an Adams Bashforth 4 method to solve an Initial Value Problem for an ODE and I am testing it against the test equation: $y'=\lambda y$ with $y(0)=1$ with the exact solution: $y(t)=...
7
votes
2answers
91 views

Algebraic multigrid for complex valued matrices

Assume one uses the classical AMG with Ruge-Stuben coarsening and direct interpolation for solving real valued problems. How can this approach be recycled to also solve complex valued problems like ...
1
vote
1answer
67 views

Adam Bashforth 4 method: how to determine starting values and stil keep the the order of accuracy

I am using an Adam Bashforth 4 method to solve an IVP problem so I need other numerical method to estimate the first 3 values. I am very much interested in finding a way to estimate the first 3 values ...
6
votes
2answers
248 views

Is Highams' computation of mean worth the price?

In Accuracy and Stability of Numerical Algorithms, equation 1.6a, Higham gives the following update formula for the mean: $$ M_{1} := x_1, \quad M_{k+1} := M_{k} + \frac{x_k - M_k}{k} $$ Ok, one ...
4
votes
2answers
102 views

SLATEC Routine Computes Givens Rotation in Unexpected Way

Some Background I am working on a C++ translation of a SLATEC routine, R1UPDT, which performs a Givens rotation: $$r = \frac{1.0}{\sqrt{a^2 + b^2}}$$ Usually, ...
2
votes
3answers
214 views

How does a stiff equation solver work?

I am trying to understand how stiff differential equations are solved. For instance the equation, $$\frac{\partial y}{\partial t} = \alpha\frac{\partial ^2 y}{\partial z^2}$$ can be solved using ...
3
votes
1answer
153 views

Derivatives of Approximate Matrix inverses

I am cross posting this question to the mathermatics stack exchange. please find it either at this link, https://math.stackexchange.com/q/2952989/430980, or below: I have a question concerning the ...
0
votes
1answer
100 views

calculating integral for an ODE system

I have an ODE system defining a mathematical model of a biological system, say $$ \frac{da}{dt}=f_1(a,b,\ldots,z,p)\\ \frac{db}{dt}=f_2(a,b,\ldots,z,p)\\ \cdots\\ \frac{dz}{dt}=f_n(a,b,\ldots,z,p) $$ ...
2
votes
1answer
113 views

Poorly conditioned, easily evaluated sum for unit testing

I am looking for examples of poorly conditioned sums which can rapidly be evaluated, for the purposes of unit testing. I'm currently using the series representation for $\ln(2)$: $$ \sum_{n=1}^{\...
3
votes
2answers
234 views

Generate high n quantum harmonic oscillator states numerically

How can I generate the higher $n$ quantum harmonic oscillator wavefunction (in position space) numerically? Here, higher means around $n=500$, or say $n=2000$, where $n$ is the $n$th oscillator ...
1
vote
1answer
105 views

Relation between conjugate gradient method and finite elements method

What is difference beetwen this two method? Are these methods far from each other or are these methods complement each other? Could you take an example?
4
votes
1answer
183 views

Numerically find the nearest positive semi definite matrix to a symmetric matrix

I have a symmetric matrix $M$ which I want to numerically project onto the positive semi definite cone. To do so, I decompose it into $M = QDQ^T$ and transform all negative eigenvalues to zero. (...
0
votes
1answer
63 views

Approximation of ODE solution using Taylor series methods

This is my first post on here, so please excuse mistakes if any. I am trying to plot out the difference between two ODE solvers based on Taylor series: 1st order acccurate: $x(t_0 + h) = x(t_0) + ...
4
votes
2answers
99 views

Analytical convergent sequence and numerical divergent sequence

Is it possible to construct a sequence that converges in theory but when computed numerically with a computer program is diverging. I feel that today our computer programs doesn't allow such ...
3
votes
1answer
205 views

Derivation of backward differentiation formulas(BDF)

I have been reading upon numerical techniques that are used to solve stiff ordinary differential equations. From the description given here, I could follow the steps till equation (5). I am finding ...
1
vote
1answer
183 views

Line integral along the edge of an isoparametrically mapped quadrilateral

I need to integrate a function along the edge of a quadrilateral (boundary integral). For example, the function is $f(x,y)=x^3+y^3$, the quadrilateral coordinates are $(0,0),(2,-1),(3,2),(1,3)$ and ...
3
votes
2answers
717 views

C# implementation of the gamma function that produces correct answers at positive integer inputs?

I need a C# implementation of the gamma function that produces correct exact answers at positive integer inputs. I took a look at MathNet.Numerics Meta.Numerics. In both cases, if you calculate ...
0
votes
1answer
44 views

Time iteration no longer smooth after using scaled units

I have a time iteration function looked on a 2D surface like this. Since the numbers wee very small i.e. hbar=6.6260700404e-34./(2*pi), my professor told me to use our own "scaled unites" during the ...
3
votes
0answers
69 views

Library for solving multidimensional (n > 3) hyperbolic PDE systems

Does there exist a library (in any programming language) for solving (numerically) systems of multidimensional first-order linear PDEs in the form $$\mathbf{u}_{t}+\hat{A}(\mathbf{x})\mathbf{u}_{\...
3
votes
1answer
346 views

Golub-Kahan-Lanczos Bidiagonalization Procedure implementation doesn't produce bidiagonal matrix

I'm trying to implement the aforementioned procedure using this website as a reference. At the end of the page the algorithm is described as follows: I think I've mapped the given algorithm to code ...
1
vote
0answers
68 views

Implementing boundary condition

I'm studying the transport of species A in the blood vessels, $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$ At x=0, I want to use the ...
1
vote
1answer
44 views

Wrong results for $2$ stage multistep method $y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$

I need to fix a code to utilise the $2$ stage multistep method : $$y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$$ Since this is an implicit method, I used a Newton-Raphson ...
0
votes
1answer
293 views

Crank-Nicolson method and mixed derivatives

I am curious if anyone had literature references or knowledge on how to apply the Crank-Nicolson (with approximate factorization) to the $$ \nabla \cdot (\nu (\nabla \mathbf{u} + \left(\nabla \mathbf{...
4
votes
0answers
76 views

Solution of constrained system of ODEs

Can someone point me in a direction to solve this kind of integral constrained system of ODEs. \begin{align} &\int_0^{1/2}\dot{y}^2(t)=p\\ &2\lambda_1\ddot{y}(t)+\pi cos(\pi y(t))=0\\ &y(...
1
vote
2answers
42 views

Determine conditions on parameters (for consistency) on RK method $y_{n+1} = y_n + ha_1f(t_n,y_n) + ha_2f(t_n + b_1h, y_n + b_2hf(t_n,y_n))$

I'm asked to find the conditions on the coefficients $a_1,a_2,b_1,b_2$ in the RK method $$y_{n+1} = y_n + ha_1f(t_n,y_n) + ha_2f(t_n + b_1h, y_n + b_2hf(t_n,y_n))$$ such that is consistent of (a) ...
0
votes
1answer
64 views

Numerical solution of non-linear first order partial differential equation (HJB)

I am trying to solve a simple optimal control problem using the Hamilton-Jacobi-Bellman equation, numerically in Python. This is proving to be rather difficult as I end up having to solve the ...
1
vote
1answer
181 views

Operation count for GMRES

One can use GMRES as it is, but there is also a version of GMRES called k-step restarted GMRES, which is used for large matrices, where $k$ is some fixed number of steps after which we take a new $x_0$...
4
votes
0answers
79 views

Comparing sum of floating points

I am currently working on a numerical algorithm involving a lot of floating point arithmetic, involving some badly conditioned problem sets. I am using the relation $|x - y| / (\max(|x|, |y|, 1)) \...
5
votes
1answer
134 views

Numerical solution of two coupled nonlinear eigenvalue problems

I would like to numerically solve the following system of coupled nonlinear differential equations: $$ -\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a + \left( g_a |...
5
votes
1answer
236 views

Numerical methods for solving a mixed type nonlinear PDE

What type of numerical methods are there to solve PDE of the sorts of: $$\begin{align} &f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))\\ &u(x,0)=G_1(x)\\ &\frac{\partial u(x,0)}...
3
votes
2answers
673 views

Implementing no-flux boundary condition reaction-diffusion PDE

I'm having trouble figuring out how to implement boundary conditions for this problem: \begin{align} \frac{\partial n}{\partial t} &= D_n\nabla^2n - \nabla\cdot\left(\frac{\chi}{1+\alpha c}n\nabla ...
3
votes
2answers
163 views

Asymptotic error of forward Euler

I'm trying to understand the asymptotic error behaviour of forward Euler (finite difference method), as timesteps are decreased (refined), so I feel trust in the method of manufactured solutions (MMS)....