# Questions tagged [numerics]

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### Shallow water equations: boundary conditions for sub- and super-critical flow

This question is (sort of) a continuation of this previously asked question. I am wondering about, in general, how we construct well-posed boundary conditions (both continous and numerical) for flow ...
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### Advice for a topic in a seminar

I am a master student in Mathematics, and I have to prepare a seminar for a course in mathematical methods for applied sciences. I have a good background in numerical analysis for ODEs, PDEs and hence ...
74 views

### Changing variables in integral to avoid infinity

I want to write a code in Fortran to solve this integral numerically: $$\int_{1095}^\infty \frac{dx}{x\sqrt{(x+644.153)(4.17 \cdot 10^{-5} x+0.145)}}$$ What is the best method for it? I tried Monte-...
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### Reinch's modification to the Clenshaw recurrence gives no improvement

The classical Clenshaw recurrence (see Algorithm 3.1 here) is less accurate as $x\to \pm 1$. So Reich proposed a modification to it, which is discussed as Algorithm 3.2 as well as by Oliver. While ...
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### Unable to achieve semi-linear running time in computation of continuant

I am trying to compute the continuant of a list of numbers $a_0, a_1,...,a_n$, defined by the recursion relation: $K_{n+1} = a_{n+1} K_n + K_{n-1}$ and $K_0 = 1$ (see Wikipedia). I am trying to use ...
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### Convection equation expanded in Legendre polynomials

In some areas of physics, for example radiation transport in plasmas, it is beneficial to re-express the convection equation in terms of a Legendre polynomial expansion. This results in a set of ...
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### Coupled pdes of the first order

May question is about possible approaches to solve the following system $$\begin{array}{rcl} \nabla{n}&=&n\,\mathbf{E},\\ \nabla\cdot\mathbf{E}&=&1-n, \end{array}$$ in general with ...
2k views

### How to Run MPI-3.0 in shared memory mode like OpenMP

I am parallelizing code to numerically solve a 5 Dimensional population balance model. Currently I have a very good MPICH2 parallelized code in FORTRAN but as we increase parameter values the arrays ...
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### Tackling multiscale problem in numerical simulation

In a dusty plasma system there are more than one component with different masses, i.e, electrons, ions,neutrals and dust grains. Accordingly, there are more than one temporal and spatial scales ...
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### Time complexity of numerical finite differences

I have a function $f:\mathbb R^N\to \mathbb R$ and I would like to compute all the partial derivatives of $f$ w.r.t. the $N$ input. What is the computational complexity using the (ones-sided) finite ...
216 views

### Numerical flux and source term in FVM (Burger's like equation)

I'm trying to solve the following equation with FVM $$u_t + f(u)_x = g(u)$$ where $g$ is some smooth function of $u$ and $f(u) = \frac{u^2}{2}$. This is really similar to Burger's equation, except ...
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### What space-time points should a known coefficient function be evaluated at when using the Lax-Friedrichs scheme to solve the transport equation?

For a scalar quantity $u = u(x, t)$, I'm considering the transport equation \begin{align} u_t + au_x &= 0, \qquad{x\in[0, L], \ t\in[0, T]} \\ u(0, t) &= u_{\text{in}}, \\ u(x, 0) &= f(x). ...
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### Compute mass matrix in vibrations problem by using finite element method

I have to compute the mass matrix of a Hexahedral mesh. There are 3 methods to compute mass matrix. I'm interested in one method which consists of dividing the mass of the element by the number of ...
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### Which optimization method can be used to do the following?

I've the following system of equations for studying information flow in the below graph, $$\frac{d \phi}{dt} = -M^TDM\phi + \text{noise effects} \hspace{1cm} (1)$$ Here, M is the incidence ...
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### Showing stability of numerical scheme: Decaying norm implies stability?

I have a little trouble formulating my question since I am not really sure what conclusions I am supposed to draw from the results I have obtained. I am sorry for the long problem formulation below, ...
139 views

### Mass conservation for hyperbolic relaxation problem

I have solved numerically the following system: \begin{cases} \partial_t{u} + \partial_x{v} = 0 \\ \partial_t{v} + \frac{1}{\varepsilon^2}\partial_x{u} = -\frac{1}{\varepsilon^2}(v-f(u)) \end{cases} ...
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### Using linear regression to find the ideal point given a set of trajectory's data

I have a set of points in 2D obtained from a pendular movement with some noise. I want to determine where is equilibrium point ($x_0$, $y_0$) from which the rope is fixed. There are at least two ...
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### Testing the time dependent Schrodinger Equation with an analytical solution?

I am numerically solving the Schrodinger Equation in 1D first and in higher dimension later, but I want to know the convergence rate of my numerical solver in different grid size and numerical methods....
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### Gradient-jump penalty term in FEM

I am slightly confused regarding the meaning of the $i-th$ gradient-jump term $[\nabla \phi_i]$ in the context of finite element methods, used in the assembly of the stiffness matrix (an example with <...
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### Evaluating integral $F(r_2) = \frac{1}{r_2^n} \int_0^{r_2} r_1^n f(r_1)\ \mathrm{d}r_1$ without growing instability

I have the following expression to be numerically integrated in a vector-based library (e.g. numpy, MATLAB, etc), $$F(r_2) = \frac{1}{r_2^n} \int_0^{r_2} r_1^n f(r_1)\ \mathrm{d}r_1,$$ where $n$ is ...
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### The velocity Verlet method and variable time steps

Does the velocity Verlet handle variable time steps? I found controversial statements about it. In the paper Skeel, R. D., "Variable Step Size Destabilizes the Stömer/Leapfrog/Verlet Method", BIT ...
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### Slope limiting for discontinuous Galerkin (DG) method

I had a question regarding the implementation of the TVB limiter for the RKDG method by Cockburn. I have seen that some implementations of the DG method use normalized Legendre polynomials such that ...
59 views

### Conjugacy in Non-linear Conjugate Gradient Descent

In linear conjugate gradient method, our goal is to solve the system of linear equations $$Ax = b$$ where A is a symmetric positive definite matrix, and that is equivalent to finding the minimizer of ...
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### How to compute the mean 2D slice of a 3D set of data in MPI

I have a 3D set of data v(i,j,k), and I want to compute the mean 2D slice vmean(i,j) summing up the ...
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### Type of Rosenbrock method by its coefficients

A Fortran code that solves stiff PDE systems contains the following arrays of Rosenbrock-Wanner method coefficients: ...
508 views

### What is the difference between Abaqus and Calculix contact input?

I would like to say first that am new at using Calculix. I'm using Abaqus/CAE to create a cup deep drawing simulation and everything worked perfectly but my objective is to run the same exact ...
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### pdepe or Crank-Nicolson? How much is pdepe good?

I am beginner in MATLAB and similar. I sow and discussed with my professors doing simulations some times: they wrote down a lot of calculus, most of them using Crank-Nicolson Method and so implement ...
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### Complex differentiation of linear solvers

I have a linear system $$Ax=b$$ which I'm solving approximately, and I need to take the frechet derivative of x with respect to z. Were I solving the problem exactly (either analytically or to machine ...
I am solving the steady, two-dimensional adjoint Euler equations, $$A_x^T \partial_x \Psi + A_y^T \partial_y \Psi = 0$$, where $A_x = \partial F_x/\partial U$ and $A_y= \partial F_y/\partial U$ are ...