# Questions tagged [runge-kutta]

The Runge–Kutta methods are a set of numerical methods for ordinary differential equations for the approximation of their solutions.

140 questions
Filter by
Sorted by
Tagged with
34 views

### Stabilized Many Stage Runge-Kutta methods instead of Local/Multirate Time Stepping

Locally refined meshes are often inevitable for accurate, yet feasible computations. In the context of time-dependent PDEs, however, this comes at the cost that (due to the CFL condition) reducing the ...
• 671
34 views

### What is the ERRCON parameter in rkqs?

Ive take a course in computational physics and was asked to implement some numerical methods to solve ODES. I was reading up on the algorithms described in the textbook: NUMERICAL RECIPES IN FORTRAN. ...
• 101
1 vote
74 views

### Solving 1D Advection Equation Using Midpoint-Rule and Finite-Differences

I want to solve the advection equation ($v_0 \in \mathbb R$) $$\frac{\partial f}{\partial t} + v_0 \frac{\partial f}{\partial x} = 0$$ using 2nd order Runge Kutta like the midpoint rule for the time ...
• 241
199 views

### Why is my Runge-Kutta 4 solution to the 1-D advection equation decaying so quickly?

I am trying to numerically solve the advection equation $y_t + y_x = 0$ using a the "classical" Runge-Kutta 4 explicit timestepping method, along with a left-hand finite difference ...
• 23
1 vote
76 views

### Direct integration of 2D Euler Equations with Runge Kutta shows oscillating Courant-Friedrichs-Lewy coefficient. Stiff or Bug?

By writing the direct integration of the 2D Euler Equations in a wide and short box where the fluid enters and exits through the horizontal faces using the Runge Kutta O(4) method I have found that ...
• 113
963 views

### Why is RK45 used as the "default" method for non-stiff ODEs rather than a multistep one?

From what I read, the "default" go-to method for non-stiff ODEs is the Dormand-Prince Runge-Kutta pair; for instance, in Matlab docs, "Most of the time, ...
• 9,170
140 views

### Jacobian matrix cutoff in ODE solver

I am studying an implementation of a 3rd semi-implicit Runge Kutta method (siRK3) from the book by Villadsen & Michelsen (1978), Solution of differential equation models by polynomial ...
• 192
515 views

### N-body problem with differents solvers (RK2, RK4, Euler symplectic, Stormer-Verlet) : planets drift to infinity

I'm trying to write an integrator for the 2 and 3-body problem. I choose to start from a generalisation to N-body problem so I can just pass my bodies to the same integrator in the two cases. I'm ...
89 views

### Divergence on wave equation simulation

I'm currenly working on my own PDE solver for non-linear simulations in python. I've done succesfully simulations for KdV and Fisher's equation, but now I'm playing with second order derivatives in ...
165 views

### Perturbation problem using Runge-Kutta 4

I'm trying to evaluate the perturbations magnitude between 2 body orbiting a central one in three dimensions. In order to do this I need to have an estimate of the error, which I did using Richardson ...
• 101
435 views

I am interested in numerical solutions of a linear, time-dependent ODE of the form $$\dot y = A(t)y - Ry,$$ A good model is the following problem in $\mathbb R^2$: $$A(t) = \begin{bmatrix}0 & -\... 3 votes 2 answers 260 views ### Using the BDF and RK4 methods to solve this coupled system of ODEs in C++ I'm trying to solve a system of ODEs using the BDF order 4 method. I find the first 3 points using RK4, then for the implicit part of the BDF, I use Newton-Raphson iteration. Unfortunately my solution ... 2 votes 1 answer 126 views ### What is the derivation of the values of a1, a2, p1 and 11 in the Second Order Runge Kutta Method? So currently I am studying about the Runge Kutta Second Order Method used to estimate first order ordinary differential equations. The following show the formulas.$$ y_{i+1} = y_i + (a_1k_1+a_2k_2)h $... • 121 0 votes 0 answers 246 views ### Python evaluating a second order ODE with RK4 Pasted below is my python code. It is a 4th order runge kutta that evaluates the 2nd order ode: y'' +4y'+2y=0 with initial conditions y(0)=1, y'(0)=3. I need help fixing it. When I run my code, my ... 3 votes 1 answer 150 views ### How to get a more accurate cancelation I shall try to get to the point, so let me know if there is something left and you need more details. I am solving a couple of equations that are not coupled explicitly, but their corresponding ... 1 vote 1 answer 630 views ### DOP853 integration method is missing (SciPy) I was checking some integration methods provided by SciPy, in which the DOP853 should be included according to the webpage (https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.... 1 vote 1 answer 204 views ### Coupled second-order differential equations using runge kutta 45 As a little summer project I have tried to make a ballistic calculator for when I play football, (following an example from a book), just to learn some numerical methods while doing so. My problem is ... • 33 1 vote 1 answer 82 views ### How is the final result calculated in RK-Dopri(4,5)? I have found a toy implementation of RK-Dopri(4,5), written in Python. I am concerned however, about line 118: y = y + h * (b1*K1+b3*K3+b4*K4+b5*K5+b6*K6) Has the ... • 649 1 vote 0 answers 84 views ### RK4-method starts oscillating above certain input parameters I am trying to solve an equation of the following type $$\partial_zE(z)=-c_0J$$ with $$J=c_1\beta E^3(z)$$ using the boost::odeint-framework and a fixed time stepper, with$c_0$,$c_1$and$\beta$... • 543 0 votes 1 answer 142 views ### Method of Lines Runge-Kutta nonlinear stability and behavior I have a system of 4 nonlinear 1st-order PDEs. I want to solve them numerically by method of lines, first discretizing space. This leads to the system of$N\times 4$coupled ODEs. $$\mathbf{u}_{i} =... 0 votes 1 answer 193 views ### Two RK4 method in one program I want to solve this integral using RK4 by coding in Fortran:$$R=∫1/a(t) dt → dR/dt=1/a(t) =f(t)$$Initial point: t=0 (or a=0.001) and R=0 And I have to get a(t) by solving another ... • 11 3 votes 1 answer 101 views ### Time independent Runge Kutta integration of SDE I am trying to compare the result of numerical integration of time independent Runge_Kutta, github page for stochastic differential equations with the analytical solution. True answer match the ... • 131 1 vote 3 answers 179 views ### Runge-Kutta method for an ODE with initial value which is root of denominator I wrote a code in Fortran to solve this differential equation using RK4 method:$$ \frac{dy}{dx}=A\sqrt{\frac{B}{y}+\frac{C}{y^2}} $$A, B, and C are some known constants. The problem is that ... • 11 0 votes 3 answers 115 views ### Changing variables in integral to avoid infinity I want to write a code in Fortran to solve this integral numerically:$$\int_{1095}^\infty \frac{dx}{x\sqrt{(x+644.153)(4.17 \cdot 10^{-5} x+0.145)}}$$What is the best method for it? I tried Monte-... 0 votes 0 answers 968 views ### Solving nonlinear pendulum using Runge-Kutta 4 for smaller steps I am trying to solve nonlinear pendulum using 4th order Runge-Kutta method for limits between a=0.0 to b=110 seconds and simulated the results to observe the pendulum movement. But when I increase the ... • 276 2 votes 1 answer 81 views ### Adaptive Runge-Kutta for Stochastic (Projected) Gross-Pitaevskii Equation I am using the XMDS library for solving the stochastic (projected) Gross-Pitaevskii equation$$i \hbar \partial \Phi\left(\mathbf{r},t\right)_t=\hat{\mathcal{P}}\left\{(1-i \gamma)\left(\hat{H}_{\... • 21 2 votes 1 answer 551 views ### Solving ODEs with nonlinear constraints I'm trying to solve an ODE problem. Let's say$\mathbf{x}(t)$represents the position of a particle at time$t$, and$\mathbf{u}(\mathbf{x},t)$is a velocity field defined in Cartesian coordinates on ... • 143 1 vote 0 answers 92 views ### Shooting Method with RK4 and Thermal Radiation I am attempting to numerically solve the following problem. I decompose it into a system of two first order ODEs and then solve via the shooting method. I use the fourth order Runge-Kutta (RK4) method ... • 131 0 votes 3 answers 608 views ### Solving coupled ODEs using Runge-Kutta method I want to solve the following sets of$n$coupled equations. Initial values of$x_{n}(t)$and$p_{n}(t)$are specified. The problem is, I have an 1D lattice where every particle is bound with ... • 119 3 votes 1 answer 131 views ### Type of Rosenbrock method by its coefficients A Fortran code that solves stiff PDE systems contains the following arrays of Rosenbrock-Wanner method coefficients: ... • 216 1 vote 0 answers 1k views ### Numerically solving a partial differential equation in python with Runge Kutta 4 I'm supposed to solve the following partial differential equation in python using Runge-Kutta 4 method in time. $$\frac{\partial}{\partial t}v(y,t)=Lv(t,y)$$ where$L$is the following linear ... • 19 3 votes 1 answer 380 views ### The Formula of Explicit Runge-Kutta Fourteen order I need an explicit Runge-Kutta 14th order formula. If you know about some reference that discusses at least 10th order (or higher, since I'm looking for the 14th) of Runge-Kutta and there is ... • 141 0 votes 1 answer 532 views ### Solving differential equation in Python with discretized variable coefficients I am trying to solve a differential equation with discretized variable coefficients which are calculated from a time serie. In this case the Runge-Kutta step size is fixed by the frequency in the time ... 6 votes 2 answers 1k views ### Runge-Kutta in the presence of an attractor Suppose you are solving a system of equations numerically that possesses an attractor (no matter the initial conditions set, all the different solutions will approach a specific set of values that ... 2 votes 0 answers 176 views ### What exactly is the cause(s) of blow-up for too-large step size in a method like RK4? I have been working on creating a few home-made numerical methods, and I am using them to visualize text-book problems from my Strogatz dynamics textbook. It feels like a good way to learn numerical ... • 183 2 votes 1 answer 614 views ### Runge-Kutta fourth order method. Integrating backwards I am using a Runge-Kutta fourth order method to solve numerically the usual equation of motion of a background scalar field in curved spacetime with a quartic potential:$\phi^{''}=-3\left(1+\frac{H^{... 1 vote
120 views

### How to Break Coupled ODEs down to first order for Runge-Kutta

My question might seem a bit simple. I am trying to solve a system of ODEs using Runge-Kutta method. I am having difficulty breaking down the equations into a system of first order ones required ...
1 vote
258 views

### Runge-Kutta for PID and system in separate calculations without filter

I need to calculate a closed-loop system in Python; specifically, obtain the PID response and then use the output to obtain the system response sample-by-sample with my own loop. For this, I am ...
79 views

### Verifying convergence of a stationary solution to a PDE with the Runge-Kutta method

I am numerically solving a nonlinear wave PDE using the Runge-Kutta method, and I know the solution I am looking for is constant in time, but I do not know the solution. What is a good way of ...
1 vote
78 views

### Using Kutta Merson on NLS

I'm trying to use the Kutta-Merson to get the same results as in the book Solitons, Nonlinear Evolution Equations and Inverse Scattering - M. J. Ablowitz - pg 140 The author propose using the Kutta-...
394 views

### Langevin equation in 4th order Runge-Kutta

I'm trying to figure out how to translate a piece of code from Velocity Verlet to Runge-Kutta, while treating the time step dependence of the thermal noise correctly. The Langevin equation for my ...
67 views

### Reverse automatic differentiation and integration

In Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control and more Sanz Serna writes: It is well known that the reverse mode of differentiation implies ...
72 views

### Computing Trajectory Equations of Kerr Geodesics

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab. The trajectories look like: I implemented the equations and solved it with the standard Runge-...
510 views

### Is there any explicit symplectic Runge-Kutta method?

As far as I know, all the symplectic Runge-Kutta methods are implicit which need to solve non-linear equations during the calculation. Is there any explicit method? If not, why?
1k views

### Dormand–Prince 5(4): How to update the stepsize and make accept/reject decision?

https://en.wikipedia.org/wiki/Dormand–Prince_method I want to implement the Dormand-Prince 4(5) version to solve Initial Value problems. Using regular notation I have $A$ matrix and the $c,b,\hat{b}$ ...
• 255
556 views

### Is there a database/website with Butcher tableaus?

I have started investigating in mostly Runge Kutta and Runge Kutta Nyström methods and there one of the only differences between the methods of the same type is their Butcher tableu. For the most ...
182 views

### How to implement adaptive step size Runge-Kutta Cash-Karp?

Trying to implement an adaptive step size Runge-Kutta Cash-Karp but failing with this error: ...
• 131
1 vote