All Questions
Tagged with euler-method or eulers-method
205 questions
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Is there an integration scheme for PDE? Like Euler method
I wonder whether PDE can be solved by a numerical scheme like Euler method. I think it is not quite feasible, since ODE has an explicit formula u' = f(u,t) which means we clearly know where and how ...
- 11
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98
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Proof that the semi-implicit Euler method is symplectic using $\det (J)=1$
Connected to this question about proving that the semi-implicit Euler method is symplectic.
The usual way is to show that the Jacobian matrix satisfies the symplectic condition:
$$J^T \eta J= \eta$$
...
- 190
2
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45
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implicit Euler-Maryuma method and the adjoint algorithm
I'm reading the paper Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem. In section 2.4, they write:
We can symmetrize an algorithm by ...
- 14.3k
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22
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Error estimate for consistency error (Exponential Euler Method)
Let $ D \in \mathbb{R}^n $ be a domain (open and connected), $ B \in \mathbb{R}^{n \times n} $ and $g \in C^1(D,\mathbb{R}) $ We consider the initial value problem:
$$ y' = By + g(y), \quad y(0)=y_0 \...
- 55
-1
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1
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57
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Linear Beamforming - Euler math
I'm reading slides regarding array sensor beamforming, but I don't understand some steps .
"If we have an array of N equally spaced sensors, the array output is the sum of the individual sensor ...
- 1
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81
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Why is the error in a Taylor series sometimes written as a big O and sometimes a small O
I'm trying to use Euler's method for the first time, and in its derivation our class book used the little oh notation to refer to the quadratic term when you have
$y(t_1) = y(t_0) + hy'(t_0) + \frac{y'...
0
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105
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Is this IVP ill-posed?
I'm being asked to (i) use Euler's method in this question to approximate the solutions of an IVP, (ii) as well as to give its theoretical error bound. The IVP is defined as follows:
$y'=\frac{1+t}{1+...
- 371
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1
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124
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Closed form numeric solutions for the harmonic oscillator via explicit and implicit Euler method
I am currently trying to apply the explicit and implicit Euler method to the harmonic oscillator
$y'' + y = 0, y(0)=0, y'(0)=1$.
I have reduced the ODE to first order via
$y'=v, v'=-y, y(0)=0, v(0)=1$....
- 175
0
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2
answers
180
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Understanding the Runge-Kutta 2nd order method [closed]
The question is in the "coding" context but I'd say it's more math related than python related. I have 2 main questions:
I've always learnt that the Taylor expression is for a function that ...
- 318
1
vote
1
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106
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Solve advection numerically
I try to solve the velocity advection equation numerically.
Advecting a scalar quantity like temperature works very well when using a fully implicit upwind-scheme based euler approach.
I have tried to ...
2
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0
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95
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Convergence of Euler-Maruyama scheme in total variation, uniformly over time
Consider the SDE
$$X(t) = X(0) + \int_0^t f(X(s))ds+ \int_0^t g(X(s-))dZ(s) \quad t\in [0,T]$$
where $f: \mathbb{R}^d \to \mathbb{R}^d$, $g: \mathbb{R}^d \to \mathbb{R}^d$ are Lipschitz functions, and ...
- 153
0
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1
answer
222
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Golden Spiral How to plot trajectory of a point on a circle?
I have a simple question. Have searched everywhere on the net and cant seem to find anything that answers this. See the attached image. The image describes how the horizontal line ( diameter ) is ...
- 31
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2
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140
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Weird error behavior when implementing Euler's method for solving ODEs in Python
I am solving this particular ODE:
$$
\begin{cases}
u'(t)=-2\,tu^2/20, \quad t \in [0,\sqrt{20}]\\
u(0) = 1
\end{cases}
$$
which the analytical solution is given:
$$
u(t) = \frac{1}{1 + t^2/20}.
$$
...
- 11
1
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1
answer
154
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Transformation of a complex function into a real function via Euler's formula?
I am new as a member but I'll try my best to make it as understandable and easy to follow as possible. I recently had to find a general solution to this homogeneous differential equation:
\begin{...
5
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225
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Reference for Shooting Method
Consider the following setup. We have a second order boundary value problem:
$$\dfrac{d^2y}{dx^2}=F(x,y,dy/dx);\qquad y(x_0)=y_0,\quad y(x_f)=y_f.$$
A numerical approach is to almost first write as ...
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