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NASA NTRS1993

Implicit multigrid Euler solutions with symmetric Total-Variation-Diminishing dissipation

A symmetric Total-Variation-Diminishing (TVD) formulation of the numerical dissipation terms has been incorporated into a diagonalized alternating direction implicit multigrid algorithm to solve the Euler equations of inviscid compressible flow. The new treatment of the dissipation makes is possible to capture both very strong and very weak shocks, virtually without oscillation for the steady flows of interest here. In addition, the TVD constraint fixes one of the two previously arbitrary constants in the formulation of the dissipation, and results in both converged solutions and convergence rates which are relatively insensitive to the choice of the remaining dissipation parameter.

Caughey, David A.↗

NASA NTRS1993

A cosmological hydrodynamic code based on the total variation diminishing scheme

An explicitly second-order finite difference code based on a total variation diminishing scheme for self-gravitating cosmological hydrodynamic systems is described. The code was developed to follow correctly the adiabatic changes of extremely supersonic preshock flows with a Mach number larger than 100 as well as very strong shocks. In calculations of the formation of a 1D Zel'dovich pancake, an energy accuracy of 1 percent is obtained for 32 cells per unit wavelength, and the accuracy reaches 0.01 percent as the number of cells approach 1024. It is shown that shocks are well resolved and separate cleanly the hot, dense, collapsed peaks from the cold, low-density, expanding voids.

Ryu, Dongsu↗

NASA NTRS1986

Convenient total variation diminishing conditions for nonlinear difference schemes

Convenient conditions for nonlinear difference schemes to be total-variation diminishing (TVD) are reviewed. It is shown that such schemes share the TVD property, provided their numerical fluxes meet a certain positivity condition at extrema values but can be arbitrary otherwise. The conditions are invariant under different incremental representations of the nonlinear schemes, and thus provide a simplified generalization of the TVD conditions due to Harten and others.

Tadmor, Eitan↗

NASA NTRS1985

Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations

The novel implicit and unconditionally stable, high resolution Total Variation Diminishing (TVD) scheme whose application to steady state calculations is presently examined is a member of a one-parameter family of implicit, second-order accurate systems developed by Harten (1983) for the computation of weak solutions for one-dimensional hyperbolic conservation laws. The scheme will not generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments for a quasi-one-dimensional nozzle problem show that the experimentally determined stability limit correlates exactly with the theoretical stability limit for the nonlinear scalar hyberbolic conservation laws.

Yee, H. C.↗

NASA NTRS1983

Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations

The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme.

Yee, H. C.↗

NASA NTRS1983

Implicit total variation diminishing (TVD) schemes for steady-state calculations

Yee, H. C.↗

NASA NTRS1993

Assessment of Total Variation Diminishing schemes in combustion instability problems

Numerical simulations of acoustic waves in a shear layer and in an idealized combustion chamber using high resolution Total Variation Diminishing (TVD) schemes has been carried out to study the effects of inherent scheme dissipation and dispersion errors of this class of problems. The numerical results are compared against available exact solutions to quantify these errors. Several popular TVD limiters widely used in the Computational Fluid Dynamics (CFD) community have been assessed. Osher-Chakravarthy limiters are modified so that they can be used in explicit schemes. Among all the limiters investigated, Osher-Chakravarthy third-order limiter is identified as having performed the best. It is also found that all TVD schemes have exceptionally small dispersive errors.

Wang, Z. J.↗

NASA NTRS1983

Application of second-order-accurate Total Variation Diminishing (TVD) schemes to the Euler equations in general geometries

A one-parameter family of explicit and implicit second-order-accurate, entropy satisfying, total variation diminishing (TVD) schemes was developed by Harten. These TVD schemes were the property of not generating spurious oscillations for one-dimensional nonlinear scalar hyperbolic conservation laws and constant coefficient hyperbolic systems. Application of these methods to one- and two-dimensional fluid flows containing shocks (in Cartesian coordinates) yields highly accurate nonoscillatory numerical solutions. The goal of this work is to expand these methods to the multidimensional Euler equations in generalized coordinate systems. Some numerical results of shock waves impinging on cylindrical bodies are compared with MacCormack's method.

Yee, H. C.↗

DOE OSTI2022

Block-structured, equal-workload, multi-grid-nesting interface for the Boussinesq wave model FUNWAVE-TVD (Total Variation Diminishing)

Abstract. We describe the development of a block-structured, equal-CPU-load (central processing unit), multi-grid-nesting interface for the Boussinesq wave model FUNWAVE-TVD (Fully Nonlinear Boussinesq Wave Model with Total Variation Diminishing Solver). The new model framework does not interfere with the core solver, and thus the core program, FUNWAVE-TVD, is still a standalone model used for a single grid. The nesting interface manages the time sequencing and two-way nesting processes between the parent grid and child grid with grid refinement in a hierarchical manner. Workload balance in the MPI-based (message passing interface) parallelization is handled by an equal-load scheme. A strategy of shared array allocation is applied for data management that allows for a large number of nested grids without creating additional memory allocations. Four model tests are conducted to verify the nesting algorithm with assessments of model accuracy and the robustness in the application in modeling transoceanic tsunamis and coastal effects.

Choi, Young-Kwang↗

NASA NTRS1993

A total variation diminishing finite difference algorithm for sonic boom propagation models

It is difficult to accurately model the rise phases of sonic boom waveforms with traditional finite difference algorithms because of finite difference phase dispersion. This paper introduces the concept of a total variation diminishing (TVD) finite difference method as a tool for accurately modeling the rise phases of sonic booms. A standard second order finite difference algorithm and its TVD modified counterpart are both applied to the one-way propagation of a square pulse. The TVD method clearly outperforms the non-TVD method, showing great potential as a new computational tool in the analysis of sonic boom propagation.

Sparrow, Victor W.↗

NASA NTRS1988

Total Variation Diminishing (TVD) schemes of uniform accuracy

Explicit second-order accurate finite-difference schemes for the approximation of hyperbolic conservation laws are presented. These schemes are nonlinear even for the constant coefficient case. They are based on first-order upwind schemes. Their accuracy is enhanced by locally replacing the first-order one-sided differences with either second-order one-sided differences or central differences or a blend thereof. The appropriate local difference stencils are selected such that they give TVD schemes of uniform second-order accuracy in the scalar, or linear systems, case. Like conventional TVD schemes, the new schemes avoid a Gibbs phenomenon at discontinuities of the solution, but they do not switch back to first-order accuracy, in the sense of truncation error, at extrema of the solution. The performance of the new schemes is demonstrated in several numerical tests.

Hartwich, PETER-M.↗

NASA NTRS1995

Explicit and implicit compact high-resolution shock-capturing methods for multidimensional Euler equations 1: Formulation

Two classes of explicit compact high-resolution shock-capturing methods for the multidimensional compressible Euler equations for fluid dynamics are constructed. Some of these schemes can be fourth-order accurate away from discontinuities. For the semi-discrete case their shock-capturing properties are of the total variation diminishing (TVD), total variation bounded (TVB), total variation diminishing in the mean (TVDM), essentially nonoscillatory (ENO), or positive type of scheme for 1-D scalar hyperbolic conservation laws and are positive schemes in more than one dimension. These fourth-order schemes require the same grid stencil as their second-order non-compact cousins. One class does not require the standard matrix inversion or a special numerical boundary condition treatment associated with typical compact schemes. Due to the construction, these schemes can be viewed as approximations to genuinely multidimensional schemes in the sense that they might produce less distortion in spherical type shocks and are more accurate in vortex type flows than schemes based purely on one-dimensional extensions. However, one class has a more desirable high-resolution shock-capturing property and a smaller operation count in 3-D than the other class. The extension of these schemes to coupled nonlinear systems can be accomplished using the Roe approximate Riemann solver, the generalized Steger and Warming flux-vector splitting or the van Leer type flux-vector splitting. Modification to existing high-resolution second- or third-order non-compact shock-capturing computer codes is minimal. High-resolution shock-capturing properties can also be achieved via a variant of the second-order Lax-Friedrichs numerical flux without the use of Riemann solvers for coupled nonlinear systems with comparable operations count to their classical shock-capturing counterparts. The simplest extension to viscous flows can be achieved by using the standard fourth-order compact or non-compact formula for the viscous terms.

Yee, H. C.↗

NASA NTRS1987

Efficient implementation of essentially non-oscillatory shock capturing schemes

In the computation of discontinuous solutions of hyperbolic conservation laws, TVD (total-variation-diminishing), TVB (total-variation-bounded) and the recently developed ENO (essentially non-oscillatory) schemes have proven to be very useful. In this paper two improvements are discussed: a simple TVD Runge-Kutta type time discretization, and an ENO construction procedure based on fluxes rather than on cell averages. These improvements simplify the schemes considerably -- especially for multi-dimensional problems or problems with forcing terms. Preliminary numerical results are also given.

Shu, Chi-Wang↗

NASA NTRS1988

Efficient implementation of essentially non-oscillatory shock-capturing schemes

In the computation of discontinuous solutions of hyperbolic conservation laws, TVD (total-variation-diminishing), TVB (total-variation-bounded) and the recently developed ENO (essentially non-oscillatory) schemes have proven to be very useful. In this paper two improvements are discussed: a simple TVD Runge-Kutta type time discretization, and an ENO construction procedure based on fluxes rather than on cell averages. These improvements simplify the schemes considerably - especially for multi-dimensional problems or problems with forcing terms. Preliminary numerical results are also given.

Shu, Chi-Wang↗

NASA NTRS2001

A High-Order Finite Spectral Volume Method for Conservation Laws on Unstructured Grids

A time accurate, high-order, conservative, yet efficient method named Finite Spectral Volume (FSV) is developed for conservation laws on unstructured grids. The concept of a 'spectral volume' is introduced to achieve high-order accuracy in an efficient manner similar to spectral element and multi-domain spectral methods. In addition, each spectral volume is further sub-divided into control volumes (CVs), and cell-averaged data from these control volumes is used to reconstruct a high-order approximation in the spectral volume. Riemann solvers are used to compute the fluxes at spectral volume boundaries. Then cell-averaged state variables in the control volumes are updated independently. Furthermore, TVD (Total Variation Diminishing) and TVB (Total Variation Bounded) limiters are introduced in the FSV method to remove/reduce spurious oscillations near discontinuities. A very desirable feature of the FSV method is that the reconstruction is carried out only once, and analytically, and is the same for all cells of the same type, and that the reconstruction stencil is always non-singular, in contrast to the memory and CPU-intensive reconstruction in a high-order finite volume (FV) method. Discussions are made concerning why the FSV method is significantly more efficient than high-order finite volume and the Discontinuous Galerkin (DG) methods. Fundamental properties of the FSV method are studied and high-order accuracy is demonstrated for several model problems with and without discontinuities.

Wang, Z. J.↗

NASA NTRS2002

High-Order Spectral Volume Method for 2D Euler Equations

The Spectral Volume (SV) method is extended to the 2D Euler equations. The focus of this paper is to study the performance of the SV method on multidimensional non-linear systems. Implementation details including total variation diminishing (TVD) and total variation bounded (TVB) limiters are presented. Solutions with both smooth features and discontinuities are utilized to demonstrate the overall capability of the SV method.

Wang, Z. J.↗

NASA NTRS1985

On the convergence of difference approximations to scalar conservation laws

A unified treatment of explicit in time, two level, second order resolution, total variation diminishing, approximations to scalar conservation laws are presented. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced and results in terms of the latter are obtained. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first order accurate in general. Convergence for total variation diminishing-second order resolution schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

Osher, S.↗

DOE OSTI2025

Stability analysis of the Eulerian–Lagrangian finite volume methods for nonlinear hyperbolic equations in one space dimension

In this paper, we construct a novel Eulerian–Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine–Hugoniot junmp condition. Unfortunately, to avoid the intersection of the partition lines, the time step sizes are still limited. To fix this gap, we detect effective troubled cells (ETCs) and carefully design the influence region of each ETC, within which the partitioned space-time regions are merged together to form a new one. Then with the new partition of the space-time domain, we theoretically prove that the proposed first-order scheme with Euler forward time discretization is total-variation-diminishing and maximum-principle-preserving with at least twice larger time step constraints than the classical first order Eulerian method for Burgers’ equation. Numerical experiments verify the optimality of the designed time step sizes.

97 MATHEMATICS AND COMPUTING↗