TVD Methods Research Articles

GRaM-X: a new GPU-accelerated dynamical spacetime GRMHD code for Exascale computing with the Einstein Toolkit

We present GRaM-X (General Relativistic accelerated Magnetohydrodynamics on AMReX), a new GPU-accelerated dynamical-spacetime general relativistic magnetohydrodynamics (GRMHD) code which extends the GRMHD capability of Einstein Toolkit to GPU-based exascale systems. GRaM-X supports 3D adaptive mesh refinement (AMR) on GPUs via a new AMR driver for the Einstein Toolkit called CarpetX which in turn leverages AMReX, an AMR library developed for use by the United States DOE’s Exascale Computing Project. We use the Z4c formalism to evolve the Einstein equations and the Valencia formulation to evolve the equations of GRMHD. GRaM-X supports both analytic as well as tabulated equations of state. We implement TVD and WENO reconstruction methods as well as the HLLE Riemann solver. We test the accuracy of the code using a range of tests on static spacetime, e.g. 1D magnetohydrodynamics shocktubes, the 2D magnetic rotor and a cylindrical explosion, as well as on dynamical spacetimes, i.e. the oscillations of a 3D Tolman-Oppenheimer-Volkhof star. We find excellent agreement with analytic results and results of other codes reported in literature. We also perform scaling tests and find that GRaM-X shows a weak scaling efficiency of ∼40%–50% on 2304 nodes (13824 NVIDIA V100 GPUs) with respect to single-node performance on OLCF’s supercomputer Summit.

GPU implementation of Explicit and Implicit Eulerian methods with TVD schemes for solving 2D solute transport in heterogeneous flows

In this work we present an efficient implementation of Eulerian TVD methods. We apply parallelization strategies based entirely on GPU for the solution of the 2D transport equation in heterogeneous porous media. Additionally, a parallel strategy is proposed for the generation of exponentially correlated lognormally distributed permeability fields in GPU. The programs are developed using C++/CUDA. The implemented methods are used to solve advective dominant problems, in a context of Monte Carlo type simulations to numerically determine the longitudinal and transversal macrodispersion coefficients averaging over 100 simulations for permeability fields for a large range of variances. The following types of transport are considered for testing: pure advection, advection-diffusion and advection-dispersion. The performance in terms of the computation time of explicit and implicit methods are compared. We show that the implemented algorithms allow to efficiently solve problems in computational domains of up to 134.5 million cells in a single GPU.

Sparse-Denoising Methods for Extracting Desaturation Transients in Cerebral Oxygenation Signals of Preterm Infants.

Preterm infants are at high risk of developing brain injury in the first days of life as a consequence of poor cerebral oxygen delivery. Near-infrared spectroscopy (NIRS) is an established technology developed to monitor regional tissue oxygenation. Detailed waveform analysis of the cerebral NIRS signal could improve the clinical utility of this method in accurately predicting brain injury. Frequent transient cerebral oxygen desaturations are commonly observed in extremely preterm infants, yet their clinical significance remains unclear. The aim of this study was to examine and compare the performance of two distinct approaches in isolating and extracting transient deflections within NIRS signals. We optimized three different simultaneous low-pass filtering and total variation denoising (LPF-TVD) methods and compared their performance with a recently proposed method that uses singular-spectrum analysis and the discrete cosine transform (SSA-DCT). Parameters for the LPF-TVD methods were optimized over a grid search using synthetic NIRS-like signals. The SSA-DCT method was modified with a post-processing procedure to increase sparsity in the extracted components. Our analysis, using a synthetic NIRS-like dataset, showed that a LPF-TVD method outperformed the modified SSA-DCT method: median mean-squared error of 0.97 (95% CI: 0.86 to 1.07) was lower for the LPF-TVD method compared to the modified SSA-DCT method of 1.48 (95% CI: 1.33 to 1.63), P<0.001. The dual low-pass filter and total variation denoising methods are considerably more computational efficient, by 3 to 4 orders of magnitude, than the SSA-DCT method. More research is needed to examine the efficacy of these methods in extracting oxygen desaturation in real NIRS signals.Clinical relevance- Early and precise identification of abnormal brain oxygenation in premature infants would enable clinicians to employ therapeutic strategies that seek to prevent brain injury and long-term morbidity in this vulnerable population.

High order nonlinear filter methods for subsonic turbulence simulation with stochastic forcing

Numerical simulations of forced turbulence in compressible fluids are challenging due to the multi-scale nature of the problem and conflicting requirements for numerical methods to accurately resolve the small scales and, at the same time, to handle shock waves and other discontinuities without generating spurious oscillations. Minimizing nonlinear instability and aliasing error while maintaining high accuracy before the simulation reaches the statistically stationary stage is computationally intensive. The goal of this work is to employ an efficient class of high order finite difference nonlinear filter methods for subsonic turbulence simulation with stochastic forcing. The 3D Euler equations for subsonic turbulence with temporally varying stochastic forcing at rms Mach numbers up to 0.6 are numerically solved using the Strang operator splitting of the homogeneous part of the Euler equations and the forcing terms. It was shown in Yee et al. (2013) that the Strang operator splitting is more stable than solving the full Euler equation with the source term included. The spatially seventh-order nonlinear filter methods with adaptive dissipation control developed by Yee & Sjögreen (2007, 2011) are used to solve the homogeneous system and an ODE solver is used to solve the forcing source terms. The nonlinear filter method includes a full time step of a spatially eighth-order central base method, using a third-order TVD Runge-Kutta time integration. The solution computed with the central base method is then nonlinearly filtered by an adaptive flow sensor and the dissipative portion of a seventh-order WENO with the Roe Riemann solver. In order to improve nonlinear stability of the base method without added numerical dissipation, the central base method discretizes the skew-symmetric split form of the inviscid flux derivatives. Both Ducros et al. and Kennedy-Gruber skew-symmetric split forms are tested. Numerical stability, computational efficiency, and effective spectral bandwidth of the nonlinear filter schemes are compared with those of second-order TVD and fifth- and seventh-order WENO methods. It is shown that the nonlinear filter method for this application is substantially more efficient, accurate and yields a superior spectral bandwidth compared to the standard TVD and WENO methods. The nonlinear filter method also demonstrates robust long-time integration for moderately compressible, statistically stationary turbulence with large-scale solenoidal forcing, including small-scale quantities such as enstrophy and mean-square dilatation.

Comparative study of high order methods for subsonic turbulence simulation with stochastic forcing

A class of spatially seventh-order nonlinear filter methods with adaptive dissipation control developed by Yee & Sjögreen [1, 2] is tested on three-dimensional subsonic turbulence simulations with stochastic forcing. The Euler equations are solved using the Strang operator splitting of the homogeneous part of the equations and the stochastic forcing term, with an ODE solver used to integrate the latter. Both Ducros et al. and Kennedy-Gruber skew-symmetric split formulations of the inviscid flux derivatives are considered to minimize the use of numerical dissipation. The nonlinear filter methods are shown to be numerically stable for this application at least up to an rms Mach number of 0.6. The performance and accuracy of this numerical approach are compared with those of second order TVD and fifth and seventh order WENO methods. The nonlinear filter methods are shown to be substantially more computationally efficient, delivering a superior spectral bandwidth compared to the standalone TVD and WENO methods.

Computational and experimental studies of rapid free-surface granular flows around obstacles

The study of the rapid free-surface granular flows that are driven under gravitational acceleration has attracted much attention in recent years. This is not only because such flows occur in many industrial and natural examples, but also because their transport mechanisms can be observed through small-scale lab experiments, modeled with continuum theories, and simulated by computers. When granular particles rapidly propagate around obstacles, the resulting phenomena – shock waves, particle free regions or granular vacua, expansion fans and stagnation zones – are of particular theoretical and practical importance. In this paper we develop a computational method for a hydraulic-type avalanche model that is able to simulate such phenomena. It numerically solves the avalanche model over a structured grid by including the topography of the obstacles. Different finite-difference TVD methods based on the non-oscillatory central (NOC) scheme are tested in computation to compare the resolution of the shock waves when granular flows propagate against an oblique wedge. This is also involved in the choice of the limiters, but which are shown rather insensitive in such computations. A level set formulation is coupled into the governing equations to follow the evolution of the boundaries of the granular vacua. It is tested in a situation when granular particles flow around a circular cylinder, where bow shock waves, granular vacua, expansion fans and stationary zones are all captured in the computation. These results agree well and consistently with the laboratory experiments.

Positivity-preserving high-resolution schemes for systems of conservation laws

A new class of flux-limited schemes for systems of conservation laws is presented that is both high-resolution and positivity-preserving. The schemes are obtained by extending the Steger–Warming method to second-order accuracy through the use of component-wise TVD flux limiters while ensuring that the coefficients of the discretization equation are positive. A coefficient is considered positive if it has all-positive eigenvalues and has the same eigenvectors as those of the convective flux Jacobian evaluated at the corresponding node. For certain systems of conservation laws, such as the Euler equations for instance, this condition is sufficient to guarantee positivity-preservation. The method proposed is advantaged over previous positivity-preserving flux-limited schemes by being capable to capture with high resolution all wave types (including contact discontinuities, shocks, and expansion fans). Several test cases are considered in which the Euler equations in generalized curvilinear coordinates are solved in 1D, 2D, and 3D. The test cases confirm that the proposed schemes are positivity-preserving while not being significantly more dissipative than the conventional TVD methods. The schemes are written in general matrix form and can be used to solve other systems of conservation laws, as long as they are homogeneous of degree one.

Mobile and immobile fluid transport: Coupling framework

AbstractWe introduce a solver method for mobile and immobile transport regions. The motivation is driven by transport processes in porous media (e.g. waste disposal, chemical deposition processes).We analyze the coupled transport‐reaction equation with mobile and immobile areas.We apply analytical methods, such as Laplace‐transformation, and for the numerical methods we apply Godunov's scheme, see (Mat. Sb. 1959; 47:271–306; Finite Volume Methods for Hyperbolic Problems. Cambridge University Press: Cambridge, 2002).The method is based numerically on flux‐based characteristic methods and is an attractive alternative to the classical higher‐order TVD methods, see (J. Comput. Phys. 1993; 49:357–393). In this paper, we will focus on the derivation of analytical solutions for general and special solutions of the characteristic methods that are embedded in a finite‐volume method.At the end of the paper, we illustrate the higher‐order method for different benchmark problems. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Front tracking in recirculating flows: a comparison between the TVD and RCM methods in solving the VOF equation

The modeling and simulation of mold filling must include a method to capture the interface formed between the inlet fluid and the fluid that was initially in the mold. A commonly used front-capturing method in a Eulerian mesh is the volume-of-fluid (VOF) method. The VOF advection equation solution may show numerical diffusion and/or dispersion and high-order numerical schemes, such as the TVD schemes with dimensional splitting, have to be employed to discretize the convective terms. The present contribution explores the use of RCM for solution of the VOF color-function equation during mold filling with recirculating flows. The Navier-Stokes equations are solved by a finite-volume method using the SIMPLER algorithm. Filling simulations using the TVD and RCM methods are compared. RCM was able to generate diffusion-free results, sharply defining the interface, even when topological changes (generation of droplets) occur.

NONLINEAR MONOTONIZATION OF THE BABENKO SCHEME

The goal of the paper is to present and test the nonlinear monotonization of the Babenko scheme for solving 2D linear advection equation with alternating‐sign velocities. The numerical method of monotonization is based on the idea of limited artificial diffusion. There are some approaches for constructing quasi‐monotonic second order approximation schemes for solving hyperbolic systems and equations of gas dynamics: flux correction methods, the Godunov method, TVD methods and others. In particular, many authors developed the idea of TVD method. We try to use this idea to get a new quasi‐monotonic high order accuracy scheme based on the well‐known non‐monotonic Babenko scheme. The algorithm is presented for 1D problem. For testing 2D problem we use the splitting algorithm. The proposed monotonized scheme has shown the best results among all considered in the paper schemes especially for non-smooth initial profile.

Nonlinear monotonization of the Babenko scheme 1

Abstract The goal of the paper is to present and test the nonlinear monotonization of the Babenko scheme for solving 2D linear advection equation with alternating‐sign velocities. The numerical method of monotonization is based on the idea of limited artificial diffusion. There are some approaches for constructing quasi‐monotonic second order approximation schemes for solving hyperbolic systems and equations of gas dynamics: flux correction methods, the Godunov method, TVD methods and others. In particular, many authors developed the idea of TVD method. We try to use this idea to get a new quasi‐monotonic high order accuracy scheme based on the well‐known non‐monotonic Babenko scheme. The algorithm is presented for 1D problem. For testing 2D problem we use the splitting algorithm. The proposed monotonized scheme has shown the best results among all considered in the paper schemes especially for non‐smooth initial profile.

Non-linear evolution using optimal fourth-order strong-stability-preserving Runge–Kutta methods

Strong-stability-preserving (SSP) time discretization methods (also known as total-variation-diminishing or TVD methods) are popular and effective algorithms for the simulation of partial differential equations having discontinuous or shock-like solutions. Optimal SSP Runge–Kutta (SSPRK) schemes have been previously found for methods with up to five stages and up to fourth order. In this paper, we present new optimal fourth-order SSPRK schemes with mild storage requirements and up to eight stages. We find that these schemes are ultimately more efficient than the known fourth-order SSPRK schemes because the increase in the allowable time-step more than offsets the added computational expense per step. We demonstrate these efficiencies on a pair of scalar conservation laws.

HIGHER ORDER NUMERICAL METHODS EVALUATION FOR THE COMPUTATION OF ONE DIMENSIONAL FREE SURFACE SHALLOW WATER FLOWS

International Journal of Computational Engineering ScienceVol. 03, No. 01, pp. 13-55 (2002) No AccessHIGHER ORDER NUMERICAL METHODS EVALUATION FOR THE COMPUTATION OF ONE DIMENSIONAL FREE SURFACE SHALLOW WATER FLOWSA. I. DELISA. I. DELISDepartment of Applied Mathematics, University of Crete, 71409 Heraklion, Crete, Greece Search for more papers by this author https://doi.org/10.1142/S1465876302000538Cited by:3 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractSeveral explicit high resolution TVD methods and their detailed application to steady and unsteady one-dimensional free surface channel flow problems are presented. Some of these methods are widely presented in the literature. Their evaluation is based on computations performed for steady state benchmark problems with analytical solutions and for a wide range of cases, which include friction, nonuniform bed slopes, hydraulic jumps and non-prismatic channel cross sections. Results for the classical benchmark problem of an idealised dam-break are also presented. The results obtained by the various TVD methods are compared with each other and the exact solutions, providing insight into the limitations of the methods along with their relative performances in robustness and accuracy. Some subtle points of the methods and the effects of selecting different options for certain methods are emphasized.Keywords:Finite DifferencesShallow WaterTVDComputational Hydraulics Engineering FiguresReferencesRelatedDetailsCited By 3A finite volume method parallelization for the simulation of free surface shallow water flowsA.I. Delis and E.N. Mathioudakis1 Jul 2009 | Mathematics and Computers in Simulation, Vol. 79, No. 11Application of the HLL Approximate Riemann Solver to One-Dimensional Open Channel Flows with Irregular GeometryXinya Ying and Sam S. Y. Wang26 April 2012Improved implementation of the HLL approximate Riemann solver for one-dimensional open channel flowsXinya Ying and Sam S.Y. Wang26 April 2010 | Journal of Hydraulic Research, Vol. 46, No. 1 Recommended Vol. 03, No. 01 Metrics History KeywordsFinite DifferencesShallow WaterTVDComputational Hydraulics EngineeringPDF download

Linearized Formulation of the Riemann Problem for Adiabatic and Isothermal Magnetohydrodynamics

In this paper we present a linearized formulation of the Riemann problem for ideal adiabatic and isothermal magnetohydrodynamic (MHD). Roe's property U is obtained. This ensures good capturing of shocks of arbitrary strength. A parameter vector is found which gives an almost consistent mean state, where consistency is defined in the sense of Roe. The eigenvalues and eigenvectors for the linearized Riemann problem for MHD have been explicitly constructed further facilitating numerical implementation. The method has been tested in the author's RIEMANN code for numerical MHD. In a companion paper on TVD methods for MHD, where this Riemann solver was used, we have shown that excellent results can be obtained. The consistency for this formulation is illustrated by the fact that the eigenvalues have the same symmetrical distribution around the Roe-averaged x-component of the velocity that the eigenvalues for ideal MHD have around the physical x-component of the velocity. The accuracy, simplicity, and noniterative nature of the linearized formulation of the MHD Riemann problem presented here make it especially suitable for use on massively parallel processor machines.

Numerical Experiments with the Multiresolution Scheme for the Compressible Euler Equations

We investigate the performance of the multiresolution method. The method was recently proposed as a way to speed up computations of compressible flows. The method is based on truncating wavelet coefficients. The method showed good performance on one-dimensional test problems. We implement the method in two space dimensions for a more complex compressible flow computation, intended to simulate conditions under which many production CFD codes are running. Our conclusion is that we can in many cases reduce the CPU time, but that the gain in efficiency is not as large as for the one-dimensional problems. We furthermore observe that it is essential to use the adaptive multiresolution method, which mixes centered differences with TVD methods.

A comparison of TVD Lax–Wendroff methods

AbstractThis paper compares two formulations of Lax–Wendroff TVD methods. The basis of comparison is to the results of several test problems providing both qualitative and quantitative results. Results show that using an upwind biased limiter provides higher resolution of both smooth and discontinuous solutions from a lower amount of induced numerical viscosity. The conclusion is that a limiter should have as small a support as possible in order to limit its effects if high resolution is the object.

What does the entropy condition mean in traffic flow theory?

Mathematical models of freeway traffic flow do not include models of the driver's ride impulse (e.g. of the fact that drivers really start if a traffic light switches form red to green). Moreover, the occurrence of shocks leads to the necessity to deal with weak solutions of mathematical models, as far as models formulated in terms of conservation laws are concerned. But the transition from a classic conservation law differential equation to its weak formulation leads to a loss of uniqueness of the solution. This also happes in gas dynamics where an entropy condition was additionally introduced in order to pick out the physically true solution. In this paper, it will be pointed out that this entropy condition can also be used in traffic flow theory as a uniqueness criterion, and that it is - strangely enough - the missing mathematical model of the ride impulse. These ideas are exemplified in case of the Lighthill-Whitham model which so seems to become again an up-to-date model that can numerically be treated by means of very effective numerical procedures such as TVD methods.