The ability to employ the use of numerical algorithms in science, and thusly decision making processes, is dependent on the validity of those models and the level of trust that may be placed in them. The verification and quantification of error that may be present in computational models is thus an important aspect of computational physics.  A differential system with physical dissipative mechanisms is solved using a high order finite difference method. The results of these simulations are compared to a robust, well-verified and validated, finite element code.

We seek to explore the mathematical physics implications of numerical operators. Most often, mathematical physics and numerical physics are developed independently and numerical results that are produced are post priori assigned physical significance. A motivation for an analysis comparing physical and numerical dissipation is that the true limit of zero dissipation is never recovered due to computational and physical feasibility. The implications of the use of dissipative, and other, numerical operators and how they mutate the numerical solution to the solution of a different differential system has not been explicitly explored and compared to modified analytic equations.

Prime examples of numerical dissipative methods are artificial viscosity and artificial heat conduction. Artificial heat conduction has been used since the 60s in order to ensure numerical stability. It is not as widespread as artificial viscosity, and is often used in conjunction with artificial viscosity to ensure numerical stability and ameliorate errors. In general, since its first use artificial viscosity has been used in order to smooth hydrodynamic flows and allow numerical solutions on the length scale of a mesh, as the inclusion of real viscosity is often not computationally or physically feasible due the length scales it acts on. This overestimation of the effect of viscosity misattributes numerical artifacts with a physical phenomenon.

Physical dissipative mechanisms are functions of the thermodynamic properties of a material. Many simulations performed treat fluids as inviscid which ignore physical dissipative mechanisms but use numerical dissipative mechanisms to ensure numerical stability. We seek to explore the disparity between these simulations with numerical dissipative mechanisms and simulations with physical dissipative mechanisms. This is done by exploring energy conservation, entropy production, and wall heating effects in the context of two problems: an ambient fluid, and a piston impinging on an ambient fluid.

Local and global entropy production and energy conservation are explored in the context of an ambient fluid problem. It is found that local entropy production may be negative, while the overall global entropy production still remains positive. In addition, it is noted that dissipation that is created via either a physical or a numerical operator may not be fully acknowledged or understood in traditional entropy analysis due to certain terms approximating entropy producing or dissipative terms.

It is found that wall heating errors are recovered with a physical Re number in a viscous only wall problem simulation. In addition, with the inclusion of physical heat conduction, it is found that the viscous-conductive solution corrects for the wall heating error, as was done via numerical dissipation by Noh. The entropy producing effects of viscosity and heat conduction are physical objects that are not always accounted for in traditional numerical schemes. A numerical coefficient is used to ensure numerical stability which may not be physically reflective. In addition, sometimes the numerical coefficients for these dissipative mechanisms may not be explicitly implemented in the numerical algorithm, but may arise from the numerical discretization.

Lastly, it is shown that physical and numerical viscosity may reproduce the same solutions for sufficiently viscous mediums. Implications of numerical algorithm verification are explored by considering the resolution required to capture a physical viscosity that is not so finely resolved as to be too computationally expensive or require alternative physics to accurately capture.


Exploration of the Implications of Numerical Operator Mutation of Differential Systems

Division / Sponsors: 
Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

IMECE2020 - Virtual Conference

Supporting the need among researchers worldwide for additional online and multimedia content, ASME now offers the ASME Conference Video Collection. This platform will collect and provide long-term access to conference presentation videos, encompassing all technology areas within ASME's technical conference and events program. About 30,000-40,000 engineers across all mechanical engineering and related disciplines participate in ASME conferences each year. This includes authors of approximately 6,000 to 8,000 technical papers published by ASME, the majority of whom also submit video presentations for our virtual conferences. The video collection will be available as an add-on to existing product packages or as a separate subscription.