Space-Time Conservation Element Solution Element Method
Shock Vortex Interaction
New Publications: Updated August 23, 2006
The engineering research and design requirements of today pose great
challenges in computer simulation to engineers and scientists who are called
on to analyse phenomena in continuum mechanics. The future will bring even
more daunting challenges, when increasingly complex phenomena must be
analysed with increased accuracy. Traditionally used numerical simulation
methods have evolved to their present state by repeated incremental
extensions to broaden their scope. A broadly conceived method is needed to
meet future simulation challenges.
At NASA GRC, researchers have been developing a new numerical framework for
solving conservation laws in continuum mechanics, namely, the Space-Time
Conservation Element and Solution Element Method, or the CE/SE method for
short. This method has been built from fundamentals, and not as a
modification of any previously existing method. It has been designed with
generality, simplicity, robustness and accuracy as cornerstones.
The CE/SE method has thus far been applied in the fields of computational
fluid dynamics, computational aeroacoustics and computational
electromagnetics. Computer programs based on the CE/SE method have been
developed for calculating flows in one, two and three spatial dimensions.
Numerous results have been obtained, including various shock tube problems,
the ZND detonation waves, the implosion and explosion problem, shocks over a
forward-facing step, the blast wave issuing from a nozzle, various acoustic
waves, and shock/acoustic wave interactions. The method can clearly resolve
shock/acoustic wave interactions wherein the difference of the magnitude
between acoustic wave and shock could be up to six orders. In
two-dimensional flows, the reflected shock is as crisp as the leading shock.
CE/SE schemes are currently being used for advanced application to jet and
fan noise prediction and to chemically reacting flows.
A list of key features and advantages of the CE/SE method follows.
- Space and time are treated in a unified fashion. The space-time
domain is discretized into Solution Elements (SEs), within which the
numerical approximant is a simple (linear, for example) function of space
and time. Conservation Elements (CEs) that fill the space-time domain
without overlap are also defined. Each "face" of a CE is a hypersurface in
space-time. Any point on a face of a CE belongs unambiguously to a single SE.
- The main emphasis is on solving the integral form of the conservation
law in the space-time domain, although the differential form is also
considered. As a direct statement of the integral form, conservation of
space-time flux is enforced for each CE. Because the CEs fill space-time
without overlap, and because the flux through any face of a CE is uniquely
defined, this local conservation of flux ensures that space-time flux is
conserved globally in the space-time domain. Traditional methods that
address flux conservation focus only on the conservation of spatial flux.
- The method is very simply described in terms of the geometry of the
space-time discretization. Only the simplest numerical approximation
techniques are used. No knowledge of the properties of the solution, such as
the characteristics or the shock-wave profile, are used in the construction
of schemes. Use of the properties would complicate the scheme and prevent it
from being general. Such solution properties should automatically be
manifested in the numerical solution obtained by a faithful discretization
of the conservation law.
- A staggered space-time mesh is employed, such that inviscid flux
information at each interface separating two CEs can be evaluated without
interpolation or extrapolation. In particular, no Riemann solver is needed
in calculating interfacial fluxes.
- The solution structure is not calculated through a reconstruction
procedure, as is done in modern upwind methods. Instead, the gradients of
the solution are treated as independent unknowns. For hyperbolic problems,
the gradients are not influenced by the solution properties in neighboring
SEs at the same time level. This is in full compliance with the physics of
the hyperbolic initial value problem.
- For simulation of isentropic, inviscid flow, the CE/SE method can be
used to construct explicit solvers that are non-dissipative (neutrally
stable) for all Courant numbers less than or equal to unity. Also, these
solvers are two-way time-marching schemes, i.e., each forward marching
scheme can be inverted to become a scheme for marching backward in time.
- The schemes for nonisentropic flow are constructed with the addition
of artificial dissipation. The artificial dissipation associated with the
inviscid fluxes is completely controlled by an adjustable parameter. When
the added artificial dissipation is set to zero, the inviscid scheme reduces
to the non-dissipative version, which has no inherent artificial
dissipation. Thus, the artificial dissipation can be made as small as
desired. Such a property is essential for the accurate calculation of
acoustic waves and thin boundary layers. Too much artificial dissipation
would damp or smear out such physical phenomena in the numerical solution.
- The various CE/SE schemes constructed to solve model 1D convection or
convection-diffusion equations display some remarkable properties. For
example, in various limiting cases, the principal amplification factor
obtained by a von Neumann stability analysis of the schemes coincides with
the amplification factor of various classical schemes, viz., the Leapfrog,
the Lax, the Crank-Nicolson and the DuFort-Frankel
schemes. However, the CE/SE schemes are completely distinct from each of the
- Systems of conservation laws are treated in exactly the same way as
single conservation laws, so that the discretized equation for a single law
can be symbolically transformed into that for a system by replacing scalars
with appropriate matrices. This fact is another manifestation of the
simplicity of the CE/SE method.
- The flux-based nature of the method leads to the use of flux-based
boundary conditions. The non-reflecting boundary conditions needed for
practical computations on unbounded spatial domains are remarkable for their
simplicity, needing only a few lines in the computer program. No
characteristic theory is employed in their construction. These conditions
allow even shock waves to pass out of the domain with no noticeable
reflection at the boundary. In contrast to this, non-reflecting boundary
conditions for traditional methods, based on characteristic theory, can
constitute the major part of the computer program. Furthermore, the latter
conditions are not applicable to shock waves.
- For flows in multiple spatial dimensions, no directional splitting is
employed. The CE/SE method results in genuinely multidimensional schemes.
Furthermore, identical principles are used in multiple dimensions, so that
the resulting schemes bear a remarkable resemblance to their 1D
counterparts. Furthermore, the multidimensional schemes share with their 1D
counterparts virtually identical fundamental properties.
- No global coordinate mappings are employed. The two- and
three-dimensional spatial meshes employed by the present method are built
from triangles and tetrahedrons, respectively. Triangles and tetrahedrons,
respectively, are also the simplest building blocks for two- and
three-dimensional unstructured meshes. Thus, the multidimensional schemes
can be directly applied on the unstructured meshes that offer an
efficient way to deal with complex geometries.
- The simplicity of the CE/SE schemes makes the computer programs easy
to vectorize and parallelize. With little programming effort, the programs
can deliver quick results on advanced-architecture computer systems.
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