# Questions tagged [numerics]

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### discretizing advection equation with variable wave speed + stability

I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form ...
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### Rank of a double-precision augmented matrix

Let $A$ be a matrix with real entries, and let $A_+$ be $A$ augmented by a single column. From linear algebra we know \begin{equation} \operatorname{rank}(A_+) = \operatorname{rank}(A) \hspace{10pt} ...
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I've a matrix equation $\bar z^{(r)}(k + 1) = \bar{DF} ~ \bar z^{(r)}(k)$. Now the $\bar z$'s and $\bar{DF}$ are $d \times d$ matrices. Now $\bar{DF} = \begin{pmatrix} 0 & 1 & 0 & 0 & \... 0answers 22 views ### offline Bin Packing problem with multiple size bins As per my research on stack overflow communities, This is probably known as cutting stock problem / multiple Knapsack problem (a variant of the bin packing problem) which is NP hard. here are the ... 0answers 76 views ### Remez algorithm convergence I have implemented the Remez algorithm in Python where all calculations were done with the Python mpmath library. I have noticed that sometimes the$|E_{max}|$and$|E_{min}|$do not monotonically ... 1answer 144 views ### Finite element (1D) for steady state non-linear problem I need to solve with linear finite elements the equation $$\frac{\partial }{\partial x}\Bigl(\text{sgn}(x) u \Big) +\frac{\partial}{\partial x} \Bigl[ \sqrt{u} \frac{\partial u}{\partial x} \Bigr] =0... 1answer 111 views ### Non-Linear advection diffusion with nondifferetiable advection term I'm looking at Murray's book: Mathematical biology: an introduction , first volume, pag. 404 In particular, I'm interested to solve the following PDE:$$\partial_t u = \partial_x (\text{sign}(x) u) + \... 1answer 85 views ### Ill-condioned Linear System and Gaussian Elimination Suppose that I have a linear system$Ax=b$such that$A$is ill-conditioned. Can I say that it is dangerous to find a solution with Gaussian Elimination for this system, or does there exist some class ... 0answers 64 views ### How can I practice multivariable root-finding? Recently, I've been reading up on various root-finding / optimization algorithms such as the Levenberg-Marquardt method, Gauss-Newton, Conjugate Gradient, trust-region and trust-region-dogleg. I've ... 1answer 59 views ### Imposing pressure variation instead of Dirichlet boundary conditions on Finite Element Method I always see Finite Element codes solving PDE with Dirichlet or Neumann boundary conditions. But, I have a problem now consisting of a straight cylinder with a circular base (a simple 3D tube), with ... 0answers 95 views ### Solution of Cahn-Hilliard equation I need to solve the Cahn-Hilliard equation $$\frac{\partial u}{\partial t} = \Delta(f(u) - \epsilon^2\Delta u), \hspace{.5cm}(x, t)\in \Omega\times(0, T],$$ using mixed formulation \begin{equation}\... 0answers 45 views ### Norm estimates if adjoints can't be computed Assume that you have two linear maps$A$and$V$. For a given$x$(of appropriate dimension) you can compute$Ax$numerically, and for any$y$(of appropriate dimension) you can calculate$V^Ty$... 1answer 106 views ### Accurate and efficient computation of the logarithm of the ratio of two sines For exploratory work related to special function implementations, I need to compute$\log \frac{\sin y}{\sin x} $, where$0 \le x \le y \le 2x < \frac{\pi}{2}$. Cases with$x \approx y$in ... 1answer 59 views ### Parity for artificial dissipation term in a finite-difference solution I have a doubt regarding the signal of the dissipative term in a finite difference solution for an equation of the form $$\frac{\partial u}{\partial x}+f(x)=0, u(0)=0$$ In which$f$is an odd ... 0answers 42 views ### numerical solution to pde on an ellipse Looking for advice on discretization (preferably finite difference) schemes for pdes on curves in general, but in this case it is an ellipse (so given by$(a\cos(r), b\sin(r)$). The problem is the ... 1answer 66 views ### Find quadrature points and weights I'm struggling with the following problem: What is the maximum degree of exactness that we can obtain with the following quadrature >formula $$\int_0^1 f(x)\frac{1}{\sqrt{x}}dx \approx w_0 f(x_0) +... 1answer 69 views ### Interpreting multivariable root-finding results from Matlab's fsolve algorithm Edit: So I was able to get the same value of r that's given, when coding up the sum of squares of function values directly in the script file, rather than on the Command Window. So, maybe there's a ... 2answers 62 views ### No flux Neumann boundary condition for non-stationary PDE equivalent to Dirichlet boundary? When using no flux Neumann boundary conditions (i.e. zero derivative to the normal on the boundary) in a non-stationary PDE, I don't seem to recognize the difference to using Dirichlet boundary ... 1answer 132 views ### How to use numerical integration to calculate the surface area of a superellipsoid? I am working in an application in which I need to calculate the surface area of a superellipsoid. I have read that there is no closed form solution (see here), so I am trying to compute it using ... 2answers 38 views ### Grid Independence Study Is the change in time step necessary for the grid independent study? As the CFL is based on the relation between dt and dx. In mesh independent study, only change should be mesh i.e, dx isn't it so? 0answers 26 views ### Set of integrators do not consistently solve an equation in Python I must solve the following second order differential equation: \delta \phi^{''}_{\mathbf{k}}+(3-\epsilon)\delta \phi^{'}_{\mathbf{k}}+\left(\frac{k^2}{a^2 H^2}+\frac{V_{,\phi\phi}}{H^2}-6\epsilon +4\... 1answer 65 views ### Extracting data from VTK simulations using C++ I have been given a few numerical simulations regarding fluid mixing and have been asked to extract a few parameters from them using C++. Altogether there are about 1000 VTK files per simulation, and ... 1answer 96 views ### Imposing no flux boundary conditions on variants of the Cahn-Hilliard equation using Finite Differences in Python I have been looking into simulations of phase separation in variants of the Cahn-Hilliard system and have been running into issues with implementing no flux boundary conditions on certain variants. ... 0answers 54 views ### The error in SOR algorithm suddenly falls to zero when it reaches 1e-7 range I am solving the Poisson equation for heterojunction using Fortran90. I use the SOR algorithm to arrive at the potential profile. I see the weird behavior where the error (the difference between the ... 0answers 25 views ### Abnormalities when using SOR to solve the Poisson Equation I am trying to solve the Poisson equation for Heterostructures using SOR. The equation to solve looks lik I have discretized the Poisson equation using finite difference and my code is written in ... 1answer 85 views ### Computing Series of ke^{-(x - h)^2} I asked this question on the Computer Science stack exchange (https://cs.stackexchange.com/questions/128710/faster-computation-of-ke-x-h2), but it appears that it is more appropriate in Computational ... 0answers 32 views ### Numerical simulation for a bounded process. Is slight deviation a “normal” fact? Suppose I have to numerically simulate a process \{y_t\} such that y_t\geq0 \forall t\in\mathbb{N}, with t denoting time-step. Let's suppose I use MonteCarlo with \mathscr{N} simulation ... 0answers 32 views ### Discretization formula for a system of two differential equations. “Solution to one of these is the initial condition of the other”. In which sense? Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where A, B and C are parameters ... 1answer 72 views ### interface value on the error equation https://www.jstor.org/stable/pdf/2157482.pdf, here I have a problem in last equation of (2.6) in section (2.1). When they are considering error equation on the interface \Gamma they get e_v^{(n)} = ... 1answer 84 views ### Which scheme for inhomogeneous convection-diffusion equation with highly variable coefficients? I have a 1D convection-diffusion equation \sigma_t = a(x,t) \sigma_{xx}+b(x,t)\sigma_x+f(x,t) defined on the unit interval, with nonzero Neumann boundary conditions at both ends. It should be noted ... 1answer 92 views ### Calculate stable time step DG method for advection-diffusion For stable time steps for the RKDG method for transport equations we require that$$ \Delta t \le \frac{\Delta x CFL}{(2k + 1)|\lambda|}, $$where \lambda is the eigenvalue of our conservation law ... 1answer 58 views ### Linear system with an l1-norm constraint I have a saddle-point system of the form \begin{equation} \begin{bmatrix} A & B \\ B^T & O \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} f \\ \vec{0} \end{bmatrix}, \end{... 0answers 37 views ### Euler Explict Method for solving the PDE for option prices under the Schwartz mean reverting model. Numerical Finance I have to solve the following PDE for a Call option : \partial_tV + \{ \alpha - (\mu - \lambda/ \alpha -log(S))\}S\partial_SV + 1/2 \sigma^{2}S^{2}\partial_{S}^{2}V - rV = 0 Where V(S,t) is the ... 1answer 67 views ### Accelerating convergence of a generalized continued fraction I wish to compute$$ \frac{1}{1 + \frac{1^3}{1 + \frac{2^3}{1 + \frac{3^3}{1+\cdots} } } } $$to high accuracy. To start, I tried computing$$ \frac{1}{1 + \frac{1^2}{1 + \frac{2^2}{1 + \frac{3^2}{1+\... 0answers 90 views ### Is Romberg integration method implemented as weighted function values numerically correct? I have to integrate expression f(x) * g(x) for many different functions f but just one g. I ... 1answer 142 views ### Accurate computation of Gauss-Kuzmin entropy The Gauss-Kuzmin distribution gives the probability of an integer appearing as a partial denominator in the continued fraction of a real number$x\$ as $$P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}... 0answers 27 views ### Error analysis of Modified Lentz's method In Numerical Recipes, the authors state: There is at present no rigorous analysis of error propagation in Lentz’s algorithm. This statement is now ~15 years old, so I wonder has this gap in the ... 0answers 42 views ### Numerical calculation of the Berry connection I'm doing some numerical calculations involving Hermitian matrices, and derivatives of the eigenvectors. Essentially, I have an n x n, Hermitian matrix H(x), which is dependent on some continuous ... 1answer 93 views ### Accurately Computing a Positive Vector in the Nullspace of a Matrix I'm sure this question has been asked before yet after many hours of searching I am unable to find a definitive answer. The problem at hand is solving the linear system:$$A \mathbf{x} = \mathbf{0}$$... 1answer 131 views ### Red flags for numerical computing? I've learnt the hard way that you should avoid: computing small numbers as the difference of two large numbers evaluating chaotic functions with imprecise inputs. Are there any other red flags a ... 0answers 23 views ### At what l/d ratio will a frame element start to behave as a shell element? I'm working in ETABS. There are few columns of dimension 300mmx1400mm. The height of building is 36.6 meter above ground level and the dimension of building is 26mx68m. I'm getting the time period of ... 0answers 77 views ### A parallelized GMRES solver? My application calls for solving a dense, 40,000 x 40,000, ill-conditioned linear system. The native SciPy GMRES solver with preconditioning has worked well for my application and solving a single ... 0answers 56 views ### How to construct a Fortin Operator for Crouzeix-Raviart Element? I want to prove the LBB condition for the Stokes Equations discretised by the Crouzeix-Raviart element. The continuous Stokes Equation in the weak formulation is Find u \in H_0^1(\Omega, \mathbb{R}^... 0answers 28 views ### Simulating the response of nonlinear system with stiff differential equations I want to simulate the response of a nonlinear system given in the following form:$$ \dot{x_1} = f_1(\bar{x_1})+g_1(\bar{x_1})x_2, \ x_1(0) = 0.2  \dot{x_2} = f_2(\bar{x_2})+g_2(\bar{x_2})x_3, \...
Abstract I have the next equation to find a force, for my problem: $$U=-\int \vec{m}\small{(x)}\times \vec{B}(x)dV$$ $$\vec{F}=-\nabla U$$ Considering 3-dimensional space with x,y,z coordinates, ...