Questions on the aspects of numerical methods/analysis relevant to scientific computing.
0
votes
0answers
49 views
Transient Fluid Dynamics
Eventually, I would like to numerically simulate the transient compressible flow in an axial compressor during start-up.
However, I know that this is a very challenging undertaking (to say the least).
...
11
votes
4answers
137 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 ...
4
votes
1answer
53 views
Energy Conservation
I'm working on a time integration scheme for my research. As a result, I have come across an interesting phenomenon. Somehow, the total energy of the scheme oscillates. At any given time the total ...
3
votes
0answers
31 views
Why do I get “estimated error” -1.#IND when doing BICGSTAB linear solver using ILUT perconditioner in eigen
I'm using Eigen (a C++ library for numerical linear algebra) to solve a linear equation with the the bi-conjugate gradient BICGSTAB algorithm with Incomplete LU preconditioner. However, the result ...
4
votes
1answer
115 views
generalized eigenvalue problem
I need to solve a real generalized eigenvalue problem
$Ax= \lambda Bx(*)$
A and B are calculated from equations below:
$$A=\sum_{i,j=1}^{N}W_{ij}(K_{i}-K_{j})\beta\beta^{T}(K_{i}-K_{j})^{T}$$
...
2
votes
1answer
60 views
Boundary value technique for heat equation
My heat equation is
$$
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad x \in [0,1], \quad t \in (0,0.1]
$$
with initial condition $u(x,0)=\sin(\pi x)$ and homogeneous ...
7
votes
2answers
110 views
Is the maximum/minimum principle of the heat equation maintained by the Crank-Nicolson discretization?
I'm using the Crank-Nicolson finite difference scheme to solve a 1D heat equation. I'm wondering if the maximum/minimum principle of the heat equation (i.e. that the maximum/minimum occurs at the ...
9
votes
2answers
153 views
What is pseudo time-stepping?
While reading some literature on PDE solvers I came across the term pseudo time-stepping today. It seems to be a common term, however I failed to find a good definition or an introductionary article ...
20
votes
3answers
221 views
How does one test a numerical ODE solver implementation?
I'm about to start working on a software library of numerical ODE solvers, and I'm struggling with how to formulate tests for the solver implementations. My ambition is that the library, eventually, ...
2
votes
3answers
164 views
Can anyone recommend a library in C++ which has the most efficiency in doing sparse matrix operations under Windows system
I have download sparselib++, but it seems that it can't be complied in Windows, only in Unix, I don't know. So can anyone recommend some which can be used in Visual Studio 2010? I want to do a large ...
3
votes
2answers
115 views
How does longer simulation time affect simulation results?
I have a transient flow and solute transport simulation running using a fortran code. The final solution time is 1 day. I need the output of hydraulic head for the output time of 0.5 day. I want to ...
2
votes
2answers
115 views
manufactured solutions for incompressible Navier-Stokes — how to find divergence-free velocity fields?
In the method of manufactured solutions (MMS) one postulates an exact solution, substitutes it in the equations and calculates the corresponding source term. The solution is then used for code ...
1
vote
2answers
68 views
Topics for project
I was assigned a project in my intro to computer programming class. The goal is to implement a numerical model to some engineering or physics/science problem. The course uses Java.
eg. soccer ball ...
0
votes
0answers
43 views
Discrete Fourier Transform Vs. Inverse Discrete Fourier Transform (Division by Number of Samples) [closed]
I've noticed that there are two possible notations for both DFT and IDFT.
The First One
DFT:
$$
y_i = \sum_{j = 0}^{n - 1} x_j \cdot \omega_n^{i \cdot j}, i = 0, \dots, n - 1
$$
IDFT:
$$
x_i = ...
2
votes
2answers
127 views
Problem Condition and Algorithm Stability
Consider 2 mathematical problems:
$$
f_1(x) = a - x \\
f_2(x) = e^x -1
$$
The condition number for a function is defined as follows:
$$
k(f) = \left| x \cdot \frac{f'}{f} \right|
$$
Lets analyze ...
5
votes
1answer
80 views
Method selection for numeric quadrature
Several families of methods exist for numeric quadrature. If I have a specific class of integrands how do I select the ideal method?
What are the relevant questions to ask both about the integrand ...
15
votes
2answers
207 views
Why do equi-spaced points behave badly?
Description of experiment:
In Lagrange interpolation, the exact equation is sampled at $N$ points (polynomial order $N - 1$) and it is interpolated at 101 points. Here $N$ is varied from 2 to 64. ...
11
votes
3answers
179 views
Numeric Quadrature with Derivatives
Most numerical methods for quadrature treat the integrand as a black-box function. What if we have more information? In particular, what benefit, if any, can we derive from knowing the first few ...
5
votes
0answers
117 views
How does positivity preservation fit into the implication chain from monotone to monotonicity preserving?
I know from "Numerical Methods for Conservation Laws" by Randall J. LeVeque that there is an implication chain of properties of methods for conservation laws:
monotone $\Rightarrow$ $L^1$-contractive ...
5
votes
1answer
171 views
Higher order Lax-Wendroff type scheme?
Suppose we want to solve a hyperbolic conservation law $u_t+f(u)_x=0$. I really like to use Lax-Wendroff, which reads
$u_j^{n+1} = u_j^n -\frac{\Delta t}{\Delta ...
5
votes
5answers
192 views
Evaluate the sum
I want to evaluate the sum $$\sum_{k=1}^\infty \left(\frac{i+1}{\sqrt{2}}\right)^k\cdot k^{-\alpha}$$ where $i=\sqrt{-1}$ and $\alpha\in[\frac{3}{4},1]$ with 8 digits accuracy.
If I am willing to ...
2
votes
0answers
45 views
Dissipation and symplectic manifolds
I'm working on an API for simulation of port-Hamiltonian systems. As far as I understand it, a Hamiltonian system is symplectic if it is power conserving, and so including resistive elements would ...
0
votes
2answers
168 views
complexity of flux limiter techniques
My question is not related to any particular problem, rather, I am looking at the equations of the form $$u_t+c(t,x)u_x=0$$ and attempt to solve it numerically. According to ...
0
votes
1answer
97 views
How easy is it to combine symbolic and numeric computation in Matlab?
CS Beta people:
I have been doing some multiple integrals with a combination of symbolic and numerical integration (because symbolic answers have not always been possible). I have been using ...
4
votes
1answer
52 views
Periodic BC for Multigrid in MD
I know that this question might be very specific and maybe nobody will know the answer, but this is probably the only community where I could find an answer:
So, as part of my master's project, I am ...
12
votes
1answer
215 views
can I trust this numerical triple integral from Matlab?
Computational Science people:
I originally posted this question at Math Stack Exchange and someone commented that I might get "much better" answers here:
I am a novice at numerical methods and ...
3
votes
2answers
101 views
Numerical simulation of particle in magnetic field
I'm interesting in numerically simulating (with my own code, not an off-the-shelf package) the motion of a single electrically charged particle in a magnetic field. The field will vary in time and ...
1
vote
0answers
55 views
How to compute the sum of a power series in a more robust way?
In order to compute the sum of a power series, we can use for loop, while loop or the analytic formula. I am wondering what is difference between those algorithms and how to improve the robustness of ...
8
votes
3answers
196 views
Sparse linear solver for many right-hand sides
I need to solve the same sparse linear system (300x300 to 1000x1000) with many right hand sides (300 to 1000).
In addition to this first problem, I would also like to solve different systems, but with ...
1
vote
1answer
73 views
Discrete convolution
please can I ask a bit stupid question? Let say I need to solve an equation in a form
$\frac{\partial X}{\partial t}=\sum_k M_k * X_{n-k}$
How can I do the discrete convolution numerically? I will say ...
3
votes
1answer
93 views
Numerical Methods for minimizing a Non-Differentiable Convex Function of Several Variables
I have a multi-variable convex continuous function which is not differentiable. I am interested to know about different numerical techniques, possibly also references to them, used for this.
Read ...
13
votes
1answer
149 views
Does transforming $J_0(x)\to\int\cos(x\sin\theta)$ help with numerical integration?
I've heard anecdotally that when one is trying to numerically do an integral of the form
$$\int_0^\infty f(x) J_0(x)\,\mathrm{d}x$$
with $f(x)$ smooth and well-behaved (e.g. not itself highly ...
2
votes
1answer
59 views
Introduction for (numerical) linear algebra of random variables
I am in search of an introduction into numerical linear algebra - or, at least, pure linear algebra - that treats the case when the input data are random variables.
A typical application would be to ...
2
votes
1answer
105 views
cholesky factorization of block matrices
I have a block matrix (either 2x2 blocks or 3x3 blocks) which is the covariance matrix for a joint space of two or three multivariate normal variables. ie
...
2
votes
1answer
152 views
generating a non-uniform grid with Chebyshev discretization
I often see that it is common to put "more points" in the region of interest in the computational domain of the numerical method, i.e. use non-uniform grid. The proofs are usually done for the uniform ...
2
votes
0answers
44 views
Conservative field mapping between two topologically disconnected surface meshes
Some background: the Front-Tracking method uses a triangular surface mesh to describe the boundary between two immiscible fluids. To deal with the breakup and coalescence of the fluid interface, ...
9
votes
2answers
138 views
What are the efficient, accurate algorithms for evaluation of hypergeometric functions?
I'm curious to know what good numerical algorithms exist for evaluation of the generalized hypergeometric function (or series), defined as
$${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = ...
5
votes
3answers
186 views
Evaluating sine and cosine of an integer multiple of an angle
When evaluating cylindrical harmonics, one needs to evaluate trigonometric functions $\cos(m\theta)$ and $\sin(m\theta)$, potentially for large integer $m$ and $\theta\in[-\pi,\pi]$. What is the best ...
6
votes
1answer
123 views
Hybrid spatial schemes for CFD: any downside to blending versus switching?
Aside from extra computational cost due to having to compute both fluxes over a certain region, is there any downside to blend two flux evaluations for a hybrid scheme in a finite volume method? The ...
11
votes
2answers
245 views
Numerical Integration - handling NaNs (C / Fortran)
I am dealing with a tricky integral that exhibits NaNs at certain values near zero and at the moment I am dealing with them quite crudely using an ISNAN statement which sets the integrand to zero when ...
1
vote
1answer
93 views
rate of convergence for the second order accurate method on two dimensional grid
I am solving $u_t=u_{xx}+u_{yy}$ with a solver which is locally $O(dx^2+dy^2+dt^2)$. I use the following norm for the error between the numerical vector solution and the analytical solution ...
5
votes
0answers
176 views
Numerical method for nonlinear system of algebraic equations of special type
I have a nonlinear system of algebraic equations of special kind:
$$
\begin{array}{rcl}
x_{i}+y_{i}+z_{0,1}+c_{i,1}z_{1,1} & = & d_{i,1}, \\
x_{i}^2 + y_{i}^2 + z_{0,2} + c_{i,1} ...
8
votes
1answer
141 views
Are there Improved ways of computing $p \log(p)$?
Most math libraries have a number of versions of logarithm functions.
Most of the time we assume them to be perfect, but actually quite a lot of them just offer a certain number of digits of ...
3
votes
2answers
282 views
Piecewise polynomial interpolation: Hermite vs Lagrange
I am a bit confused of the qualitative behavior of the two methods. Consider quadratic case, start by having points $x_i$, where I know the value and points $y_j$, where the values to be found. If I ...
4
votes
1answer
92 views
Computing flux of vector field numerically with regular grids
I would like to compute the flow rate (mL/s) of a pipe flow given the 3D velocity field $\mathbf{v} = (v_x, v_y, v_z)$ over the computational domain (a curved pipe). The field is represented in the ...
2
votes
1answer
62 views
Reaction-diffusion system animation
I'm using following code to generate a picture of the reaction-diffusion process:
Explicit Euler method too slow for reaction-diffusion problem
I would like to obtain an animation effect. When I ...
5
votes
2answers
235 views
Numerical Green functions for a nonlinear wave equation
I am trying to put down some code to get numerically the solution of the following PDE:
$$
\partial^2_t\phi-\partial^2_x\phi+\lambda\phi^3=\delta(x)\delta(t).
$$
Of course, there are several ...
3
votes
1answer
192 views
Helmholtz and Biharmonic equation examples with exact solution
I'm looking for examples of Helmholtz and Biharmonic equations in Cartesian co-ordinates with exact solutions, in order to compare my numerical solutions with it.
I was able to find quite a few ...
7
votes
4answers
293 views
Fourier pseudo-spectral method and numerical dissipation
Performing a direct numerical simulation of isotropic turbulence with Fourier pseudo-spectral method (Orzag & Patterson, PRL, 1972) using FFT.
For a background of the method, which is widely used ...
5
votes
1answer
84 views
Open source implementation of rational approximation to a function
I am looking for some open source implementation (any of Python, C, C++, Fortran is fine) of rational approximation to a function. Something along the article [1].
I give it a function and it gives me ...
