Questions tagged [numerics]
The numerics tag has no usage guidance.
802
questions
9
votes
3answers
4k views
Scientific computing vs numerical analysis
I'm a double major in computer science and mathematics. I love both subjects. I'm thinking in taking a graduate career, perhaps in scientific computing. What's the real difference between scientific ...
5
votes
2answers
500 views
Solving PDE with spatial and temporal derivatives on left hand side
I wish to solve an equation of the form,
$$
\frac{\partial}{\partial t} \left( \frac{\partial \phi}{\partial x} \right) = -\frac{\partial}{\partial x}(\mathcal{F})
$$
for the variable $\phi$ (e.g. ...
1
vote
1answer
189 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 ...
1
vote
0answers
84 views
How to use Wolfe-Powell step-size control in quasi-Newton method?
I'm trying to find the minimum of a function using the quasi-Newton method with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm.
But I want to change the following implementation, so that:
1) ...
6
votes
1answer
95 views
Going back in time in an initial value problem
Consider an initial value problem (IVP) $y'=f(t,y)$ with the initial value given by $y(t=0) = 0$.
If I need to find $y(t^*)$, hence finding the path for $y$ in $t \in [0,t^*]$ and $t^*<0$; is the ...
3
votes
1answer
76 views
How to show the stability of $L^2$ projection?
If $\mathcal{T}_h$ is a regular and quasi-uniform triangulation of $\Omega$, and $V_h$ is the $H^1$-conforming linear finite element space. Moreover, let $P_h$ be the $L^2$ projection to $V_h\subset H^...
3
votes
1answer
537 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 ...
2
votes
1answer
149 views
In iterative methods, are matrix decompositions considered useful for implementation?
When we study an iterative method from textbooks, for example, see the Gauss-Seidel Method, the given matrix is decomposed with suitable splittings. In the example, $A = L+U$. So we can proceed with ...
8
votes
1answer
3k views
computing turbulent energy spectrum from isotropic turbulence flow field in a box
I have my 3 dimensional velocity flow-field u, v and w at a given instant of time from DNS using pseudo-spectral method. I need to calculate the energy spectrum ( in Fourier space ) as a function of ...
4
votes
1answer
197 views
Accurate way of getting the square root inverse of a positive definite symmetric matrix
What is the most accurate algorithm to get the square root inverse of a positive definite symmetric matrix? I am not looking as much for efficiency, though using quadruple precision computation is out ...
0
votes
0answers
30 views
Fit exponential convergence
I'm working with a numerical algorithm whose output $y$ asymptotically approaches a certain unknown value $a$.
I expect an exponential convergence, i.e. the data $y$ given by my algorithm should be ...
3
votes
1answer
102 views
Stability region of explicit midpoint method
Consider the explicit midpoint method, i.e
$$y_{n+1}-y_{n-1} = 2hf(y_n).$$
I'm asked to apply this method to the linear test equation, $f(y_n) = \lambda y_n,$ in order to find the method's stability ...
-2
votes
1answer
85 views
How to : numerical integration by quadrature in C language / remove NaN
What I wanna solve it the problem following
( by quadrature method )
I want to get two arrays of data ( z & tau )
from z[0], tau[0] to z[2249], tau[2249].
Since the integrand diverges at z=0.9, ...
1
vote
0answers
59 views
Growing error from a smooth initial condition for Fisher KPP equation
I'm studying the Fisker-KPP equation on the line (and in $]0, 100[$ numerically):
$$
\partial_t u = \Delta_{xx} u + u(1-u)
$$
I notice a behavior I don't understand with a smooth initial condition $...
0
votes
1answer
156 views
Solving $n$ coupled equations numerically in Matlab
I would like to solve the following equations simultaneously and numerically for all $X, Y, Z, W$ where i = 1:Nw, j = 1:Nl, k = 1:K.
$W_\text{net1}$, $W_\text{net2}...
Ertxiem - reinstate Monica
2501 gold badge2 silver badges9 bronze badges
1
vote
1answer
110 views
Dirichlet to Neumann Operator
EDIT: I am trying to specify my Question. Also I am not going to clearify which spaces I use, because I am only interested in the basic idea.
I am looking at a standard elliptic second order PDE:
\...
2
votes
0answers
44 views
How to determine the order of convergence of the Euler-Maruyama method?
This question is originally posted in Quant.StackExchange but has been unanswered for some time so I ask in here.
To make this simple let us consider the Geometric Brownian Motions (GBM).
My ...
1
vote
0answers
262 views
Numerical integral with a weakly singular kernel with a satisfactory precision
I am working on a numerical method for time fractional PDE. One problem is that I must compute a numerical integral of the following form:
$$
\begin{equation}
\int_0^{t_m} (t_m-s)^{-\beta}f(s)ds
\end{...
7
votes
1answer
143 views
Element-wise thresholding a low-rank matrix in O(n) time?
Define the element-wise thresholding operator $T_\tau(\cdot)$ with threshold $\tau$ as
$$ [T_\tau(X)]_{i,j} = \begin{cases} X_{i,j} &\mbox{if } |X_{i,j}| \ge \tau, \\
0 & \mbox{if } |X_{i,j}|...
1
vote
1answer
654 views
Proper boundary conditions for potential flow around cylinder
I am computing the stationary, incompressible, inviscid and irrotational flow around a circular cylinder using a discretization in general coordinates.
I derived a PDE and proper boundary conditions ...
1
vote
0answers
49 views
Singular Spectrum Analysis Explanation
I need you to help me understand the Singular Spectrum Analysis algorithm. I already read a lot of articles about the subject but they never answered my questions like what is the mathematical reason ...
2
votes
2answers
86 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
66 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
66 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
83 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
45 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
91 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
42 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
483 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
78 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
124 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
62 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 ...
10
votes
5answers
4k 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
104 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
176 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
70 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:
$$\int_\Omega \partial_t u=\...
3
votes
1answer
137 views
What does the exponential function mean in numerical ODE solving formulas?
I'm trying to read papers on numerical ODE algorithms and I always seem to stumble upon huge amounts of exponentials multiplied by each other. For example in New families of symplectic splitting ...
10
votes
1answer
158 views
Are there shortcuts for numerically approximating systems of ordinary differential equations when autonomous?
Existing algorithms for solving ODEs handle functions $\frac{dy}{dt} = f(y, t)$, where $y \in \mathbb R^n$. But in many physical systems, the differential equation is autonomous, so $\frac{dy}{dt} = f(...
0
votes
1answer
86 views
Actual global error vs theoretical global error: How to combine theory with practice
I have implemented an Adams Bashforth 4 method to solve an Initial Value Problem for an ODE and I am testing it against the test equation:
$y'=\lambda y$ with $y(0)=1$ with the exact solution: $y(t)=...
7
votes
2answers
102 views
Algebraic multigrid for complex valued matrices
Assume one uses the classical AMG with Ruge-Stuben coarsening and direct interpolation for solving real valued problems. How can this approach be recycled to also solve complex valued problems like ...
1
vote
1answer
69 views
Adam Bashforth 4 method: how to determine starting values and stil keep the the order of accuracy
I am using an Adam Bashforth 4 method to solve an IVP problem so I need other numerical method to estimate the first 3 values. I am very much interested in finding a way to estimate the first 3 values ...
6
votes
2answers
254 views
Is Highams' computation of mean worth the price?
In Accuracy and Stability of Numerical Algorithms, equation 1.6a, Higham gives the following update formula for the mean:
$$
M_{1} := x_1, \quad
M_{k+1} := M_{k} + \frac{x_k - M_k}{k}
$$
Ok, one ...
4
votes
2answers
105 views
SLATEC Routine Computes Givens Rotation in Unexpected Way
Some Background
I am working on a C++ translation of a SLATEC routine, R1UPDT, which performs a Givens rotation:
$$r = \frac{1.0}{\sqrt{a^2 + b^2}}$$
Usually, ...
2
votes
3answers
218 views
How does a stiff equation solver work?
I am trying to understand how stiff differential equations are solved.
For instance the equation,
$$\frac{\partial y}{\partial t} = \alpha\frac{\partial ^2 y}{\partial z^2}$$
can be solved using ...
3
votes
1answer
155 views
Derivatives of Approximate Matrix inverses
I am cross posting this question to the mathermatics stack exchange. please find it either at this link, https://math.stackexchange.com/q/2952989/430980, or below:
I have a question concerning the ...
0
votes
1answer
102 views
calculating integral for an ODE system
I have an ODE system defining a mathematical model of a biological system, say
$$
\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)
$$
...
2
votes
1answer
118 views
Poorly conditioned, easily evaluated sum for unit testing
I am looking for examples of poorly conditioned sums which can rapidly be evaluated, for the purposes of unit testing.
I'm currently using the series representation for $\ln(2)$:
$$
\sum_{n=1}^{\...
3
votes
2answers
345 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
110 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?
