numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

, adaptive mesh refinement (AMR) is a method of adapting the accuracy of a solution within certain sensitive or turbulent regions of simulation, dynamically and during the time the solution is being calculated. When solutions are calculated numerically, they are often limited to pre-determined quantified grids as in the Cartesian plane which constitute the computational grid, or 'mesh'. Many problems in numerical analysis, however, do not require a uniform precision in the numerical grids used for graph plotting or computational simulation, and would be better suited if specific areas of graphs which needed precision could be refined in quantification only in the regions requiring the added precision. Adaptive mesh refinement provides such a dynamic programming environment for adapting the precision of the numerical computation based on the requirements of a computation problem in specific areas of multi-dimensional graphs which need precision while leaving the other regions of the multi-dimensional graphs at lower levels of precision and resolution. This dynamic technique of adapting computation precision to specific requirements has been accredited to Marsha Berger, Joseph Oliger, and Phillip Colella who developed an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

for dynamic gridding called ''local adaptive mesh refinement''. The use of AMR has since then proved of broad use and has been used in studying turbulence problems in hydrodynamics as well as in the study of large scale structures in astrophysics as in the Bolshoi Cosmological Simulation.

Development of adaptive mesh refinement

In a series of papers, Marsha Berger, Joseph Oliger, and Phillip Colella developed an

for dynamic gridding called ''local adaptive mesh refinement''. The algorithm begins with the entire computational

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...

covered with a coarsely resolved base-level regular

Cartesian grid A regular grid is a tessellation of ''n''-dimensional Euclidean space by congruent parallelotopes (e.g. bricks). Its opposite is irregular grid. Grids of this type appear on graph paper and may be used in finite element analysis, finite ...

. As the calculation progresses, individual grid cells are tagged for refinement, using a criterion that can either be user-supplied (for example

mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...

per cell remains constant, hence higher

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

regions are more highly resolved) or based on

Richardson extrapolation In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A^\ast = \lim_ A(h). In essence, given the value of A(h) for several values of h, ...

. All tagged cells are then refined, meaning that a finer grid is overlaid on the coarse one. After refinement, individual grid patches on a single fixed level of refinement are passed off to an

integrator An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output. Integration is an importan ...

which advances those cells in time. Finally, a correction procedure is implemented to correct the transfer along coarse-fine grid interfaces, to ensure that the amount of any conserved quantity leaving one cell exactly balances the amount entering the bordering cell. If at some point the level of refinement in a cell is greater than required, the high resolution grid may be removed and replaced with a coarser grid. This allows the user to solve problems that are completely intractable on a uniform grid; for example,

astrophysicists The following is a list of astronomers, astrophysicists and other notable people who have made contributions to the field of astronomy. They may have won major prizes or awards, developed or invented widely used techniques or technologies within a ...

have used AMR to model a collapsing giant molecular cloud core down to an effective resolution of 131,072 cells per initial cloud

radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...

, corresponding to a resolution of 10¹⁵ cells on a uniform grid. Advanced mesh refinement has been introduced via functionals. Functionals allow the ability to generate grids and provide mesh adaptation. Some advanced functionals include the Winslow and modified Liao functionals.

Applications of adaptive mesh refinement

When calculating a solution to the

shallow water equations The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). ...

, the solution (water height) might only be calculated for points every few feet apart—and one would assume that in between those points the height varies smoothly. The limiting factor to the resolution of the solution is thus the grid spacing: there will be no features of the numerical solution on scales smaller than the grid-spacing. Adaptive mesh refinement (AMR) changes the spacing of grid points, to change how accurately the solution is known in that region. In the shallow water example, the grid might in general be spaced every few feet—but it could be adaptively refined to have grid points every few inches in places where there are large waves. If the region in which higher resolution is desired remains localized over the course of the computation, then ''static mesh refinement'' can be used - in which the grid is more finely spaced in some regions than others, but maintains its shape over time. The advantages of a dynamic gridding scheme are: #Increased computational savings over a static grid approach. #Increased storage savings over a static grid approach. #Complete control of grid resolution, compared to the fixed resolution of a static grid approach, or the Lagrangian-based adaptivity of

smoothed particle hydrodynamics Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysi ...

. #Compared to pre-tuned static meshes, the adaptive approach requires less detailed a priori knowledge on the evolution of the solution. #The computational costs inherit properties of the physical system. In addition, the AMR methods have been developed and applied to a wide range of fluid mechanics problems, including two-phase flows, fluid-structure interactions, and wave energy converters.

Development of adaptive mesh refinement

Applications of adaptive mesh refinement

References

See also