An Unconditionally Stable MacCormack Method

Selle, Andrew; Fedkiw, Ronald; Kim, Byung Moon; Liu, Yingjie; Rossignac, Jarek

doi:10.1007/s10915-007-9166-4

Cited by 237 publications

(169 citation statements)

References 26 publications

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“…However, the result produces artificial viscosity which dampens vortices prematurely, and requires an iterative linear solver to solve for a pressure field to enforce incompressibility. Works aimed at mitigating or minimizing artificial diffusion include vorticity confinement [Fedkiw et al 2001] and high order advection schemes [Selle et al 2008]. To improve the performance of the iterative pressure solver, use of adaptive grids [Losasso et al 2004] and hierarchical coarse grids for projection [Lentine et al 2010] have been proposed.…”

Section: Computer Graphicsmentioning

confidence: 99%

Fluid simulation using Laplacian eigenfunctions

2012

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We present an algorithm for the simulation of incompressible fluid phenomena that is computationally efficient and leads to visually convincing simulations with far fewer degrees of freedom than existing approaches. Rather than using an Eulerian grid or Lagrangian elements, we represent vorticity and velocity using a basis of global functions defined over the entire simulation domain. We show that choosing Laplacian eigenfunctions for this basis provides benefits, including correspondence with spatial scales of vorticity and precise energy control at each scale. We perform Galerkin projection of the Navier-Stokes equations to derive a time evolution equation in the space of basis coefficients. Our method admits closed form solutions on simple domains but can also be implemented efficiently on arbitrary meshes.

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Section: Computer Graphicsmentioning

confidence: 99%

Fluid simulation using Laplacian eigenfunctions

2012

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“…[6] introduced FLIP to overcome advection dissipation. Other methods including BFECC [7], QUICK [8], MacCormack [9] suggested using higher-order space discretization schemes and higher-order time integration schemes (e.g. RungeKutta methods).…”

Section: Previous Workmentioning

confidence: 99%

View-Dependent Multiscale Fluid Simulation

Gao

Ren

et al. 2013

IEEE Trans. Visual. Comput. Graphics

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Abstract-Fluid motions are highly nonlinear and non-stationary, with turbulence occurring and developing at different length and time scales. In real-life observations, the multiscale flow generates different visual impacts depending on the distance to the viewer. We propose a new fluid simulation framework that adaptively allocates computational resources according to the human visual perception. First, a 3D empirical model decomposition scheme is developed to obtain the velocity spectrum of the turbulent flow. Then, depending on the distance to the viewer, the fluid domain is divided into a sequence of nested simulation partitions. Finally, the multiscale fluid motions revealed in the velocity spectrum are distributed non-uniformly to these view-dependent partitions, and the mixed velocity fields defined on different partitions are solved separately using different grid sizes and time steps. The fluid flow is solved at different spatial-temporal resolutions, such that higher-frequency motions closer to the viewer are solved at higher resolutions and vice versa. The new simulator better utilizes the computing power, producing visually plausible results with realistic fine-scale details in a more efficient way. It is particularly suitable for large scenes with the viewer inside the fluid domain. Also, as high-frequency fluid motions are distinguished from low-frequency motions in the simulation, the numerical dissipation is effectively reduced.Index Terms-fluid simulation, Hilbert-Huang transform, fluid velocity spectrum, view-dependent partition.

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“…It may be possible to make a second order accurate scheme using only order first order accurate interpolation stencils, as was done in the modified MacCormack scheme of [44] and the modified BFECC scheme of [6]. Another interesting approach would be to apply a second order non-conservative correction to a full conservative first order accurate scheme.…”

Section: Interpolationmentioning

confidence: 99%

“…Recently, authors have considered using semi-Lagrangian methods as building blocks in other schemes, for example [17,18,6] showed that the second order accurate BFECC method of [5] can be made unconditionally stable using the first order accurate semiLagrangian method as a building block. In addition, [44] showed that the original scheme of MacCormack [36] can be made unconditionally stable in a similar way. A notable feature of the semi-Lagrangian method is that it relieves the time step restriction.…”

Section: Introductionmentioning

confidence: 96%

An unconditionally stable fully conservative semi-Lagrangian method

Lentine

Gretarsson

Fedkiw

2011

Journal of Computational Physics

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Semi-Lagrangian methods have been around for some time, dating back at least to [3]. Researchers have worked to increase their accuracy, and these schemes have gained newfound interest with the recent widespread use of adaptive grids where the CFL-based time step restriction of the smallest cell can be overwhelming. Since these schemes are based on characteristic tracing and interpolation, they do not readily lend themselves to a fully conservative implementation. However, we propose a novel technique that applies a conservative limiter to the typical semi-Lagrangian interpolation step in order to guarantee that the amount of the conservative quantity does not increase during this advection. In addition, we propose a new second step that forward advects any of the conserved quantity that was not accounted for in the typical semi-Lagrangian advection. We show that this new scheme can be used to conserve both mass and momentum for incompressible flows. For incompressible flows, we further explore properly conserving kinetic energy during the advection step, but note that the divergence free projection results in a velocity field which is inconsistent with conservation of kinetic energy (even for inviscid flows where it should be conserved). For compressible flows, we rely on a recently proposed splitting technique that eliminates the acoustic CFL time step restriction via an incompressible-style pressure solve. Then our new method can be applied to conservatively advect mass, momentum and total energy in order to exactly conserve these quantities, and remove the remaining time step restriction based on fluid velocity that the original scheme still had.

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An Unconditionally Stable MacCormack Method

Cited by 237 publications

References 26 publications

Fluid simulation using Laplacian eigenfunctions

Fluid simulation using Laplacian eigenfunctions

View-Dependent Multiscale Fluid Simulation

An unconditionally stable fully conservative semi-Lagrangian method

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