Knut Waagan - Simulating the stars
Development of turbulent structures in a simulation of compressible flow with high Mach numbers (meaning that the gas moves faster than the sound speed). The images show mass density along a two-dimensional cut at different times. In this simulation it was critical to ensure that i) mass density and pressure remained positive numbers, ii) mass, momentum and energy was conserved, and iii) the second law of thermodynamics was properly taken into account. (Click Image for larger view)
The laws of physics are formulated as mathematical relations. For example, gases and liquids are modeled by the Navier-Stokes equations. In this model the gas is described as a continuum with the basic properties mass density, pressure and velocity, referred to as state variables. These properties are functions of time and spatial location. The theory is formulated such that if the state of the gas is known at some time, the future states should be predicted by the equations. The mathematical problem of making such a prediction is called an initial value problem.
Because of their complexity, and their importance in science and engineering, solving initial value problems is a vast area of research. In most cases the only possible solution method is to make approximate versions of the equations. These approximations either have a known solution, or they consist of only simple arithmetic operations instead of the original abstract mathematical form. Hence, we can program a computer to perform this arithmetic for us. In fact, the number of arithmetic operations needed is way too large to calculate by hand.
Since fluids are abundant in nature, the Navier-Stokes equations have a wide range of applications. One particularly challenging discipline for computations, is the simulation of astrophysical flows. Since one obviously can not just set up, for instance, a supernova in a laboratory, computer simulations take a role similar to laboratory experiments in astrophysics. Some challenging salient features of the flows in our universe are shock waves, turbulence and vast ranges of scales. Also, magnetic fields are often of crucial importance to the behaviour of astrophysical systems. A successful model in this more general context is given by the equations of magneto-hydrodynamics, which describes the interaction of ionised gases and magnetic fields.
Knut Waagan's work is centered around developing computational methods for these flows. The translation of the equations to computer code in the most straight forward manner leads to bad or useless results. Instead, great care must be taken to ensure that the computational method will reflect the actual dynamics. Accomplishing this requires both physical insight and mathematical analysis. For example, in the presence of shock waves, one needs to take into account the second law of thermodynamics (stating that heat can never be completely converted to work) to ensure that the computed energy transfer processes are physically realisable. Mathematically this means that one must augment the modeling equations with an inequality. Our computable approximation must in turn satisfy an analogous mathematical constraint. Another example has to do with the absence of magnetic monopoles in nature. It is of course desirable to have this extra constraint reflected in a computer simulation. In addition, dealing with this issue in the right way is often crucial to whether a computational method will actually work at all. His work has involved Waagan in physical topics such as the solar atmosphere, the solar wind, star formation and turbulence theory.
ASP Spotlight October 2008
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