Politecnico di Milano
Discussion
Started 28 April 2023
What is the stiffness concept in hyperbolic PDE? how to calculate it?
In the context of scalar hyperbolic partial differential equations (PDEs), the concept of "stiffness" generally refers to the behavior of the solution as the time step or grid spacing is decreased. Specifically, a PDE is said to be "stiff" if the solution changes rapidly over small time scales or distances, requiring a correspondingly small time step or grid spacing in order to accurately capture the behavior of the solution.
Stiffness is a particularly important concept in the numerical solution of hyperbolic PDEs, as these equations often exhibit sharp gradients or shocks that can be difficult to accurately capture using standard numerical methods. Stiffness can lead to numerical instability, slow convergence, and other issues that can make it difficult to obtain accurate solutions to hyperbolic PDEs. As such, developing numerical methods that can handle stiffness is an important area of research in computational mathematics and scientific computing.
I need to know this answer is correct? how to calculate is?
Can you suggest any reference please
Most recent answer
Note that there isn't a strict mathematical definition of stiffness.
In my opinion, there are two very different scenarios to consider when talking about stiffness for hyperbolic pdes, well summarized here https://en.wikipedia.org/wiki/Stiff_equation.
One is where the integration step is solely determined by the stability requirements and not by the accuracy requirements. This, in my opinion, is a largely misleading definition, because it only makes sense with respect to a numerical method. Yet, it is one that works also for a single PDE/ODE with a single eigenvalue.
Another, more convincing, one, in my opinion, is for a system of PDE/ODEs. In this case one can define the stiffness ratio, between the largest and smallest eigenvalue. Then, depending from the application, one can or not consider the problem stiff. Imagine incompressible flows where your velocity scale u is very small but the time step is still restricted, if nothing is done about it, by the speed of sound c. If you're not interested in the acoustics, that is the transient over which the pressure waves propagate, then it is clearly a stiff problem, where something you don't care about is forcing your integration step.
However, if you still have an incompressible flow (intended, again, as a flow with very small u) but you are actually interested in the transient pressure wave propagation at c, can we still say that the problem is stiff? I don't think so, because the physics you want to describe has exactly that speed. Roughly speeking, if you want to follow a wave, you need to consider its speed as part of the physics, not a numerical problem. This, obviously, doesn't exclude that there might be some clever numerical method to make things better but, again, I think the numerical method should be taken separate from the problem stiffness description.
All replies (8)
University of Campania "Luigi Vanvitelli"
You are talking about a scalar linear equation like
df/dt+u*df/dx= a*f
where the stifness depends on the coefficient a.
This topic is investigated in some Leveque publications.
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National Centre for Medium Range Weather Forecasting
Filippo Maria Denaro sir ,I appreciate your response. I'll read the Leveque book.
I need to look in to stiffness condition about the equation df/dt+u*df/dx= a*f
University of Campania "Luigi Vanvitelli"
have a look to his reference cited here
Conference Paper Accuracy and Stability Analysis of Time-Integrated Schemes f...
Islamic University of Science and Technology
In hyperbolic partial differential equations (PDEs), the stiffness concept refers to the property of the equation that causes numerical methods to require very small time steps to accurately resolve the solution. This is typically due to the presence of very different characteristic speeds in the problem, which can result in numerical instability if not properly accounted for.
To quantify the stiffness of a hyperbolic PDE, one common approach is to use the Courant-Friedrichs-Lewy (CFL) condition. This condition relates the time step size, the grid spacing, and the maximum speed of information propagation in the problem. Specifically, the CFL condition requires that the time step size be chosen small enough so that the information propagation does not exceed one grid cell per time step.
The CFL number, which is a dimensionless parameter, is defined as the product of the time step size and the maximum speed of information propagation divided by the grid spacing. A CFL number greater than one indicates that the time step is too large and numerical instability is likely to occur. Therefore, a stiffness parameter can be defined as the inverse of the maximum CFL number over all grid points. This gives a measure of the overall stiffness of the problem, with larger values indicating a stiffer problem.
In practice, the stiffness of a hyperbolic PDE can also depend on the specific numerical method being used to solve it. Therefore, it is important to consider the choice of numerical method along with the stiffness parameter when selecting appropriate time step sizes and grid resolutions for a given problem.
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Oles Honchar Dnipro National University
Prof. Ifran Nasir Wani gave a good definition for CFL number. He treats it as the stiffness criterion but it's true for numerical methods for hyperbolic PDE solving but not for PDE itself. Reynolds number - ratio velocity to viscosity together with some length (to get dimensionlessness - I'm tired of typing this word!) may be good criterion for advective-diffusion PDE (IMHO)
Politecnico di Milano
Note that there isn't a strict mathematical definition of stiffness.
In my opinion, there are two very different scenarios to consider when talking about stiffness for hyperbolic pdes, well summarized here https://en.wikipedia.org/wiki/Stiff_equation.
One is where the integration step is solely determined by the stability requirements and not by the accuracy requirements. This, in my opinion, is a largely misleading definition, because it only makes sense with respect to a numerical method. Yet, it is one that works also for a single PDE/ODE with a single eigenvalue.
Another, more convincing, one, in my opinion, is for a system of PDE/ODEs. In this case one can define the stiffness ratio, between the largest and smallest eigenvalue. Then, depending from the application, one can or not consider the problem stiff. Imagine incompressible flows where your velocity scale u is very small but the time step is still restricted, if nothing is done about it, by the speed of sound c. If you're not interested in the acoustics, that is the transient over which the pressure waves propagate, then it is clearly a stiff problem, where something you don't care about is forcing your integration step.
However, if you still have an incompressible flow (intended, again, as a flow with very small u) but you are actually interested in the transient pressure wave propagation at c, can we still say that the problem is stiff? I don't think so, because the physics you want to describe has exactly that speed. Roughly speeking, if you want to follow a wave, you need to consider its speed as part of the physics, not a numerical problem. This, obviously, doesn't exclude that there might be some clever numerical method to make things better but, again, I think the numerical method should be taken separate from the problem stiffness description.
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