Questions tagged [stiffness]

Referring to ordinary differential equations that require an extremely small timestep to guarantee numerical stability.

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Pretension a truss mechanism (using the direct stiffness method)

Currently, I'm working on a mechanical mechanism where nodes are connected via beams. This is very comparable to planar truss mechanism analysis, but in my case, the deformations are large (relative ...
Patrick's user avatar
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Why do I get an oscillatory solution when applying the implicit trapezoidal method to the linear diffusion equation?

I wish to solve the following equation, $$\frac{\partial f}{\partial t}=\frac{\partial}{\partial x}\left(D(x)\frac{\partial f}{\partial x}\right)$$ using an exponential integrator. I discretize this ...
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Exponential Integrator to solve PDE with Stiff term

I wish to solve an equation like the following, $$\frac{\partial f}{\partial t}+\frac{\partial}{\partial x}\left(A(x)f\right)=S(x,t)f$$ where $A(x,f)f$ and $S(x,t)f$ are the advection and the source ...
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Using backward and forward Euler method to solve a certain stiff ODE

When using the backward and forward Euler methods to solve a certain stiff differential equation, what criteria does one look at before drawing the conclusion that one is more stable than the other?
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Are stiffness and instability equivalent?

To the best of my knowledge, stiffness of ordinary differential equations is difficult to capture but can be roughly described as problems where explicit methods don't work while implicit ones do. ...
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Solving detailed combustion kinetics in CFD, where to start?

I have some experience solving single- and multicomponent Euler equations for modeling of gas flows, including combustible ones. The code (variations of finite-difference WENO methods) is written with ...
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ODEs that are jointly stiffer than they are individually

I am looking for an example of a certain pair of ODEs. Consider two independent ODEs $$ \frac{\partial x}{\partial t} = f(x)\ \text{and}\ \frac{\partial y}{\partial t} = g(y) $$ where $x \in \mathbb{R}...
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How to estimate stability and stiffness of a system of coupled ODEs?

I'm running into issues with Python/Julia ODE solvers requiring prohibitively small timesteps to evolve a system of 4 coupled ODEs (the order of magnitude of the state variables and time unit span ~40-...
quantumflash's user avatar
4 votes
2 answers
277 views

Backward Euler + Quasi Newton(Broyden) method fails to solve Van der Pol's equation(Stiff ODE)

The first guess is using the forward Euler approach. The first jacobian is using finite differences. Then NR method is used to solve for the next iteration and Broyden's method is used to update the ...
underdog's user avatar
2 votes
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Solving stiff ODEs: Dealing with Jacobian terms which take too long to compute with finite differences

I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
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Is there a graphical interpretation or explanation of automatic differentiation compared to numerical differentiation

I have been looking at automatic differentiation for solving differential equations lately. I understand the basic ideas of using Dual numbers and such for finding derivatives, etc. However, I feel ...
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Does non-dimensionalizing ODEs affect the stiffness of the equations?

Does non-dimensionalizing ODEs affect the stiffness of the equations? Can it improve the stability of numerical methods like ode45,ode113 in MATLAB? I am trying to solve 2 eqns. which might involve ...
Ronnie1993's user avatar
2 votes
1 answer
163 views

Second Order Time Integration for Stiff Linear System Avoiding any Explicit Step

I have a linear system $$ \dot x(t) = Ax(t), \quad x(0)=x_0 \tag{*} $$ with $A$ being Hurwitz (i.e. the solutions may oscillate but will eventually tend to zero) but really stiff. $A$ might be large, ...
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Ill-conditioned stiffness matrix

I am writting a Fem code in c++ for a 2d plane stress model. My question is regarding the assembly stiffness matrix.I noticed that some elements of the matrix are not exactly zero but insted a number ...
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scipy's solve_ivp returns erroneous results for a stiff differential equation

I'm using scipy's solve_ivp for solving a stiff differential equation. I'm using method BDF for solving the same. I have already used MATLAB's ode23s and I'm getting correct results in MATLAB. However,...
hinata exc's user avatar
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ODE forth-order very stiff equation with large errors

I’m using Mathematica home edition software to numerically solve a specific inflation equation in cosmology. The ODE equation is forth- order, non-linear, stiff. I was using the stiffness switching ...
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Unexpectedly Slow Convergence Implicit Euler

I'm solving the coupled ODE $$ \left[\begin{array}{c}x^\prime(z)\\p_x^\prime(z)\end{array}\right] = C(z)\cdot\left[\begin{array}{c}x(z)\\p_x(z)\end{array}\right] = \left[\begin{array}{cc}0& A(z)\\...
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Using nondimensionalization to solve an ode in MATLAB [duplicate]

I am trying to solve an ode that uses some extremely large numbers and some extremely small numbers, namely $$ e = 1.6\times 10^{-19}\\ E = 10^6\\ \tau = 6\times 10^{-24}\\ m = 9.1\times 10^{-31}\\ c ...
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Lax-wendroff for stiff source terms

I am interested in problems of the form $$ u_t = F(u) + S(u) $$ where $F(u) = - div(f(u))$ and $S(u)$ is a stiff source term. I am looking for any existing works which develop Lax-Wendroff type ...
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Equivalent of multiple-scale analysis for a linear ODE

I came across the method of multiple-scale analysis and was intrigued, because I am trying to solve a linear ODE with multiple characteristic timescales. When I apply the method as described, say, ...
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Integrators for Nonlinear/Stiff PDE

It was suggested I ask this question in this section. Anyway: I have a particular nonlinear PDE of the form $$ u_t(x,t)=iu_{xx}(x,t)+f(x,u(x,t)) \tag{1} $$ Where f is some nonlinear function. With ...
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What is the state of the art in solving stiff initial value problems?

I'm looking for current references on solving stiff ODEs. Most of what I know (say, BDF methods) apparently date back to the 1980's, and I feel like a lot of progress should have been made in that ...
emprice's user avatar
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3 answers
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How does a stiff equation solver work?

I am trying to understand how stiff differential equations are solved. For instance the equation, $$\frac{\partial y}{\partial t} = \alpha\frac{\partial ^2 y}{\partial z^2}$$ can be solved using ...
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Derivation of backward differentiation formulas(BDF)

I have been reading upon numerical techniques that are used to solve stiff ordinary differential equations. From the description given here, I could follow the steps till equation (5). I am finding ...
Natasha's user avatar
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Step-size selection for an Trapezoidal Method ODE solver (ode23t)

I was reading the documentation of the MatLab ODE solver ode23t, and I've seen that the trapezoidal rule is used. Moreover, the error is estimated by ...
VoB's user avatar
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Is the time step size of a Rosenbrock method for stiff systems iteratively calculated?

I have an ODE system of the general form y' = k(y)(x) + q(z)(x) x' = a(z)(x) + b(x)(x) where k,q,a and b are also dependent on the states x and y. The ...
Nirjhar Alam's user avatar
3 votes
2 answers
292 views

Failing integration with the radau5-implementation in DotNumerics, over a discontinuity

Summary We are trying to solve some models of emission of substances to the air. In these models, emission stops at a certain point. We are using the DotNumerics implementation of radau5 We had ...
R. Schreurs's user avatar
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Checking if a given differential equation is stiff

i have to decide if the following differential equation is stiff: $$y''(t)=-201y'-200y^2 + 2, \quad t\in[0,20].$$ Sadly, I don't have any solutions. So, what I did was implementing explicit and ...
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stop condition in scipy.integrate.ode for stiff system

I'm using Python scipy.integrate.ode, and I want to stop my integration at a certain condition. So I use the integrator "dopri5" and use the method "set_solout" to specify a function for my stop ...
Peter B's user avatar
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12 votes
2 answers
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Understanding the Courant–Friedrichs–Lewy condition

I understand these equations in particular can be solved easily without use of computational methods. Although right now I am concerned with trying to solve these equations using numerical integration ...
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Implementing odespy for system of PDEs

After trying to use RK4 to solve the below system of equations, it appears the output had large and fast oscillations even with an adaptive time step I incorporated using the Cash-Karp method. I am ...
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Applying Runge-Kutta to nonlinear system of PDEs

I am applying a 4th order Runge-Kutta code, using the method of lines, to solve the following: \begin{equation} \frac {\partial y_1}{\partial t} = y_2 y_3 - C_1 y_1 \end{equation} \begin{equation} \...
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How to properly implement Backwards Euler on a system of bodies

I have a system of two bodies, $b^i$ and $b^j$, each in position $\vec{p}^i = (x^i, y^i)$ and $\vec{p}^j = (x^j, y^j)$. The two are connected via spring and I'd like to know, given their states at ...
WhiteTiger's user avatar
6 votes
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490 views

Will penalty-augmented stiffness matrix cause numerical issues in eigenvalue analysis?

In the finite element method, we often construct the constraints of the system by adding penalty-function terms ( which often are many many magnitudes, up to $10^6$ order bigger than the largest ...
Graviton's user avatar
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2 votes
1 answer
239 views

Numerical methods for coupled stiff PDEs

I'm dealing with a set of nonlinear coupled PDEs that have the form: \begin{align} \frac {\partial y_1}{\partial t} &= y_2y_3 - y_1 \tag{1}\\ \frac {\partial y_2}{\partial t} &= y_1y_3 - y_2 \...
Mathews24's user avatar
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3 answers
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Solving ODE with multiple equilibriums

Consider an ODE of the form: $$ u'(t)=-\frac{1}{\varepsilon}u(u-\frac{1}{2})(u-1) $$ with the initial value $$ u(0)=u_0. $$ Here $\varepsilon>0$ is a constant. It is easy to verify that $u\equiv0$ ...
Michael's user avatar
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Stiff ODEs coupled with PDEs (computational efficiency)

I am simulating in COMSOL a system of 3 coupled PDEs (parabolic & elliptic) along with 10 stiff ODEs. In order to have the system working, I am downsizing the time step size too much to achieve ...
MBM's user avatar
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1 vote
1 answer
175 views

Applicable solvers for nonlinear coupled PDEs

I've been trying to find an applicable PDE solver for cases such as this: Although when dealing with stiff equations in the complex domain, applying existing packages has been problematic. I've ...
Mathews24's user avatar
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Stiffness emerges as number of ODEs increases

I want to solve a system of ordinary differential equations with Matlab. I need this to solve a mechanical engineering related problem. If $n$ is the number of degrees of freedom of my mechanical ...
Roberto Belotti's user avatar
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FEM, Direct Stiffness Method with a nonlinear displacement constraint in one node

i have a question about a FE problem im working on. I made a finite element model of an linear elastic block of material (double striped block) attached with a rigid connection to the environment (...
JPlanken's user avatar
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1 answer
621 views

Convection diffusion reaction equation (stiffness, solver)

I am trying to solve the CDR-Equation in 2D: $$\frac{\partial c(x,y)}{\partial t} + \nabla \cdot ( -d\nabla c(x,y) + \vec{v}(x,y) c(x,y))+ a c(x,y)=0\,,$$ with Boundary Conditions (length of square ...
Paulinchen2's user avatar
-3 votes
1 answer
222 views

How can I solve stiff equations by method of line (MOL)?

I want to solve 7 coupled equations.I use method of line(MOL) and discrete the equation in Length and radius and convert them to a system of ODEs in time.and use ode15s to solve them in MATLAB. But an ...
fatemeh's user avatar
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2 votes
0 answers
63 views

Resolving a stiff hyperbolic problem with Neumann boundary conditions

I am trying to numerically resolve the equation for an Euler-Bernoulli beam that is inextensible, unshearable, and subject to planar deformations: $$\rho I(s) \frac{\partial^2 \theta}{\partial t^2}(s,...
namu's user avatar
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16 votes
2 answers
14k views

In FEM, why is the stiffness matrix positive definite?

In FEM classes, it's usually taken for granted that the stiffness matrix is positive definite, but I just can't understand why. Could anyone give some explanation? For instance, we can consider the ...
user123's user avatar
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Spectral Collocation (or Weighted Residual) Methods to solve Stiff ODEs?

I have a system of ODEs which is (at least moderately) stiff. Consider the class of spectral collocation methods https://en.wikipedia.org/wiki/Spectral_method or the related class of weighted ...
jlperla's user avatar
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3 votes
1 answer
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Does the global stiffness matrix size depend on the number of joints or the number of elements?

When assembling all the stiffness matrices for each element together, is the final matrix size equal to the number of joints or elements?
ThrowAway's user avatar
4 votes
1 answer
769 views

What is the case of trade-off in different Runge-Kutta methods

There are so many Runge-Kutta methods, including Dormand-Prince 45, Cash-Karp 54, Fehlberge 78. Is there any comparison between them? E.g. What is each approach sacrificing? What is the general ...
ar2015's user avatar
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-1 votes
2 answers
163 views

RK4 for stiff IVP

I need a solver of stiff Inital-Value Problems (IVP) in python exploiting RK4 preferably explicit. I have been searching for past few days but could not find it. Following are my queries: Does the ...
zen's user avatar
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1 vote
0 answers
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Resources for viscous behavior in simple FEM

I am working on a simple explicit-integration lumped-mass elastic FEM code which implements CST+DKT triangles (plate+shell) and constant-strain tetrahedra (http://woodem.eu/doc/theory/membrane-element....
eudoxos's user avatar
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2 votes
2 answers
484 views

Computing element stiffness matrices with variable coefficients

I am trying to implement a simple FEM approach, using p1 triangular elements, for solving the diffusion equation with variable nodal diffusivities and I was wondering how to incorporate the variable ...
semper's user avatar
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