This tag is for requests for books, papers, and citations.

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3
votes
1answer
144 views

Numerical Methods for Solving a Fully Nonlinear Time-Dependent PDE?

Are there numerical methods of solving the following fully nonlinear time-dependent PDE: $$\nabla^2u\left(\textbf{r}(t), \dot{r}(t), t\right)=f\left(\textbf{r}(t), \dot{r}(t), t\right),$$ for $\textbf{...
8
votes
1answer
145 views

Integer vs float multiplication performance, modern CPUs

Are there benchmarks for how many multiplications of various integer types compared to floating point types can be achieved per second on modern CPUs? I'm trying to get some hint if it would be ...
0
votes
1answer
60 views

Pseudo random numbers

I am learning how to use pseudo random number generators but the instructor just told us how they work without explaining why they work. For example, can one prove that the numbers generated by LCG ...
2
votes
2answers
103 views

Why do Newton-Krylov iterations stagnate in this problem? [closed]

Consider this integro-differential heat equation taken from SciPy documentation page: $ \nabla^2 P = \alpha \left(\iint_\Omega \cosh(P)dx dy \right)^2 $ which was found in this question. In the ...
4
votes
1answer
250 views

Intro to DG Finite Element methods

I wrote a number of 1D/2D FE and FD programs as a bachelor student, but the main problem I continually came into contact with was gradient shocks related to convection/diffusion problems in convection-...
8
votes
1answer
161 views

When is it advantageous to iterate integrals numerically?

If there is an $(n+1)$-dimensional integral of the form $$ \int_{[0,1]^{n+1}} f(x, y)\,\mathrm{d}^n x \,\mathrm{d}y,$$ normally one would evaluate this using a multi-dimensional integration library ...
0
votes
1answer
177 views

Introduction to Lattice Boltzmann methods [closed]

I am trying to learn the Lattice-Boltzmann method and was looking for some good beginner resources explaining the method. I have been looking at some codes online, but have been having trouble ...
4
votes
1answer
101 views

Lid-driven Cavity benchmark in 3D. Classical paper to compare

I'm looking for a benchmark for the lid-driven cavity problem in 3D to compare the results of my code. In 2D I used: U. K. N. G. Ghia, K. N. Ghia and C. T. Shin (1982) High-Re solutions for ...
4
votes
1answer
87 views

Resources for solving mixed left and right matrix equations

I'm looking to solve a matrix equation and not sure where to start looking for resources. The equation is $$AX + XB = C\,,$$ where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{m\times m}$, $C\in\...
5
votes
1answer
66 views

Proof of CFL condition for RKDG scheme

The cfl condition for linear advection equation $$ u_t + a u_x = 0 $$ using a DG method of degree $k$ polynomials, upwind flux and an RK scheme of $k+1$ stage/accuracy is stated to be $\frac{1}{2k+1}$...
4
votes
1answer
108 views

Fourth order IMEX Runge-Kutta method

I am looking for the Butcher tableau of a fourth order accurate Runge-Kutta method with IMEX splitting. I have been reading the ''classical'' paper on the subject by Ascher, Ruuth and Spiteri as well ...
1
vote
0answers
45 views

convergence of a method

I want to show convergence of a finite element method for a higher order equation.I have a coupled equations that solved together and gives two variables as answer $[u, v]$. $$w+\Delta u=0$$ $$\...
3
votes
1answer
60 views

Strategies for controlling number of new elements in adaptive mesh refinement

I am working on adaptive techniques for solving some elliptic equations. The technique is based on residual on elements. My problem is that when I use a predefined tolerance for refining elements, the ...
6
votes
3answers
180 views

Benchmarks for Gröbner bases and polynomial system solution

In the recent question Solving system of 7 nonlinear algebraic equations symbolically, Brian Borchers experimentally confirmed that Maple can solve a polynomial system that Matlab/Mupad cannot handle. ...
1
vote
0answers
34 views

stop criterion for adaptive technique

I am working on a meshfree method that is so similar to FEM. It is neccesary for me to find an adaptive technique for that. The major problem is that I could not find any good stop criterion for that....
5
votes
3answers
170 views

Good introduction to numerical methods for MHD

I very recently started to read up about magnetohydrodynamics (MHD). While I have experience in the fluid part (both theory and numerics), my knowledge about the magneto part is very limited. At the ...
1
vote
2answers
117 views

2d Euler manufactured solutions

Where can I find manufactured solutions for the 2d Euler equations, with the complete analytical terms, including the Jacobian of the source term ?
0
votes
1answer
31 views

Reference Material

I am starting to get into scientific computing with a library called Deal.II and I was wondering what the community recommends as good source material that I can learn about scientific material. ...
18
votes
6answers
1k views

Modern C++ in scientific computing?

I am looking for books or articles, or blog-posts, or any published material in general, that address specifically the uses of C++ modern features (move semantics, the STL, iterators, lazy evaluation, ...
3
votes
2answers
118 views

Looking for reference on Streamline Upwind Petrov Galerkin finite elements for incompressible unsteady Navier-Stokes

I am looking for a relatively simple book/paper that explains the basic Streamline Upwind Petrov Galerkin (SUPG) method for solving the incompressible unsteady Navier-Stokes equations. Most of the ...
4
votes
3answers
161 views

Point inside curved finite element

I like to create interpolation functions for second order finite element meshes. For elements with straight edges all is good, but some of my elements may have curved edges as shown in the figure: ...
5
votes
1answer
59 views

Reference request: theory regarding time evolution of closed loop 2D elastic shapes?

I am interested in approximating the time evolution of 2D curves. Here's an illustration: An issue that arises when naively making this approximation as illustrated above, is that as one increases ...
1
vote
0answers
47 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
1
vote
1answer
42 views

Resources exploring the problem of “volume exclusion”?

Consider the following situation: There are two boundaries -- one is denoted using grey lines, and the other is denoted using black lines. The boundaries are numerically represented using "vertices"...
1
vote
0answers
38 views

Recommend route for research in numerical backward stochastic differential equation

I am a first year master student. My supervisor assigned numerical backward SDE as my master thesis topic and let me read a thesis from his former PHD student. After finishing reading the thesis, I ...
1
vote
3answers
98 views

Reference Suggestions for MPI

I'm learning MPI in order to use MPI with a model written in Fortran. What are some good resources (books, websites, etc.)? Introductory/beginner material, and detailed references would both be ...
3
votes
1answer
59 views

Resource recommendations for numerical methods involved in dynamical systems analysis

I am interested in learning numerical methods that specifically have to do with analyzing dynamical systems. In particular: drawing phase plane diagrams drawing phase portraits analyzing ...
4
votes
1answer
102 views

Numerical computation of $\log \int_a ^b f(x) \mathrm{d}x$ from $\log f(x)$?

I want a numerical method to evaluate: $$\log \int_a ^b f(x) \mathrm{d}x$$ when what I have is a numerical routine to evaluate $\log f(x)$. The problem is that if $f(x)$ takes very large or very ...
5
votes
1answer
106 views

Need a good reference for numerical transport phenomena

I'm a chemical engineering undergraduate and I'm currently starting to work in a theoretical transport phenomena/colloid science group. While my group has a nice code base for larger scale ...
2
votes
2answers
146 views

Looking for references on this adaptive Runge–Kutta method (GSL’s rk2)

Background For a study that is beyond the scope of this question, I applied all of GSL’s adaptive Runge–Kutta methods to a certain problem. This includes a Runge–Kutta method of 2nd and 3rd order, ...
2
votes
0answers
55 views

Resources for large-scale MILP optimization

With the advent of "big data" applications, different algorithms have to be used to efficiently solve optimization problems, even in the convex case (e.g. the recent success of stochastic gradient ...
2
votes
0answers
34 views

Adaptive plotting of two-variable functions $z=f(x,y)$ algorithm pseudocode?

I am looking for explanations of algorithms to adaptively sample a function of two variables $f(x,y)$, in a given domain $x_0\le x \le x_1$, $y_0\le y \le y_1$. Intuitively, I want to sample more ...
9
votes
3answers
248 views

Finite elements on manifold

I'd like to solve some PDEs on manifolds, say for example an elliptic equation on a sphere. Where do I start? I'd like to find something that use preexisting code/libraries in 2d , nothing so fancy (...
3
votes
3answers
156 views

Algorithms for radiation treatment planning

I have a medical physics problem - I want to maximise the dose absorbed by a brain tumour whilst minimising the dose in the rest of the brain, especially certain organs, such as the pituitary gland, ...
3
votes
1answer
75 views

How to Check a Hyper-Cube for Defects

I would greatly appreciate some help/references on solving the following problem: You are in charge of searching through a n-dimensional hyper-cube $[0,1]^n$ to make sure that it does not contain ...
7
votes
1answer
162 views

Good tutorials on how to use Butcher tables?

I tried to go to the primary sources in order to understand how to use Butcher tables to simplify the algebra I need to do when using Taylor series to find the order of accuracy of a scheme, for ...
6
votes
0answers
84 views

Reference request for numerical variational method

I have a variational problem where the unknown function is a periodic path $\gamma:[0,1)\to\mathbb{R}^2$, and the functional is $$ \int_0^1\left( \tfrac12\|\dot\gamma(s)\|^2 + \mathcal{F}[\gamma]\...
0
votes
0answers
55 views

Agent-Based Modeling

I am interested in learning about and learning how to use agent-based modeling. Specifically, I want to use agent-based modeling for economics research. Can anyone suggest resources appropriate for ...
5
votes
1answer
270 views

ENO/WENO component-wise vs characteristic-wise

Can someone give some references to understand what's the differences between a component-wise and a characteristic-wise ENO scheme? If I'm right, the characteristic variables come from the ...
-1
votes
1answer
59 views

Can you give some information for rothe method [closed]

I want to learn a numerical method for PDEs other than finite difference method. After some research on internet i have found Rothe method and it looks interesting to me. Unfortunately, i couldn't ...
1
vote
3answers
432 views

Applying the method of lines to parabolic PDEs: references and software

Could you please advise some literature about the numerical method of lines (MOL) for parabolic PDEs? It is a method of solving PDEs with discretizing only by space but not by time. A system of ODEs ...
9
votes
4answers
262 views

Fast explicit solution for $\mathbf{A}\mathbf{x} = \mathbf{b}$, $ \mathbf{b} \in \mathbf{R}^3$, low condition number

I am looking for a fast (dare I say optimal?) explicit solution the 3x3 linear real problem, $\mathbf{A}\mathbf{x} = \mathbf{b}$, $\mathbf{A} \in \mathbf{R}^{3 \times 3}, \mathbf{b} \in \mathbf{R}^{3}$...
1
vote
0answers
65 views

How to compute frank copula and its derivative accurately?

I need to fit a model using MLE with Frank copula by linking two discrete univate distribution function $u = F(x)$ and $v = F(y)$ together, and the joint distribution function is $$ \Phi(x,y) = C(F(x)...
2
votes
2answers
25 views

Runge-Kutta with all nodes at n+1 or zero weights otherwise

So, lets say for the family of the explicit Runge-Kutta methods: $$y_{n+1} = y_n + \sum_{i=1}^s b_i k_i$$ where, $$k_1 = hf(t_n, y_n)$$ $$k_2 = hf(t_n+c_2h, y_n+a_{21}k_1)$$ $$\vdots$$ $$k_s = hf(t_n+...
1
vote
1answer
116 views

Buckling reference using the FEM

I want to analyze buckling in a composite using the FEM. So far I have studied this references Zdenek P Bazant, Luigi Cedolin. Stability of Structures: Elastic, Inelastic, Fracture and Damage ...
1
vote
2answers
132 views

Reference for approximation errors in 2D and 3D by using FEM

I'm currently searching for an elaborate referece that covers most of the approximation errors for elliptic second order problems (like, for the laplacian dirichlet problem) by using finite element ...
5
votes
2answers
603 views

Looking for Runge-Kutta 8th order in C/C++

I would like to use Runge-Kutta 8th order method (89) in a celestial mechanics / astrodynamics application, written in C++, using a Windows machine. Therefore I wonder if anyone knows a good library / ...
13
votes
3answers
393 views

Is it well known that some optimization problems are equivalent to time-stepping?

Given a desired state $y_0$ and a regularization parameter $\beta \in \mathbb R$, consider the problem of finding a state $y$ and a control $u$ to minimize a functional \begin{equation} \frac{1}{2} \...
4
votes
3answers
502 views

Books on mathematical foundation of finite element methods

After reading three books about finite element method, with two of them covering also finite volume and grid generation, I found myself lost when I have to discuss these topics with library developers ...
0
votes
0answers
66 views

Lattice Boltzmann Method

I have done Molecular Dynamics Simulation and now want to venture into Lattice Boltzmann Method. What would be the best reference book/lecture notes/videos for a beginner?