查看更多>>摘要：In this paper,we study systems of conservation laws in one space dimension.We prove that for classical solutions in Sobolev spaces Hs,with s＞3/2,the data-to-solution map is not uniformly continuous.Our results apply to all nonlinear scalar conservation laws and to nonlinear hyperbolic systems of two equations.
查看更多>>摘要：This paper reviews the adaptive sparse grid discontinuous Galerkin(aSG-DG)method for computing high dimensional partial differential equations(PDEs)and its software implemen-tation.The C++software package called AdaM-DG,implementing the aSG-DG method,is available on GitHub at https://github.com/JuntaoHuang/adaptive-multiresolution-DG.The package is capable of treating a large class of high dimensional linear and nonlinear PDEs.We review the essential components of the algorithm and the functionality of the software,including the multiwavelets used,assembling of bilinear operators,fast matrix-vector product for data with hierarchical structures.We further demonstrate the performance of the package by reporting the numerical error and the CPU cost for several benchmark tests,including linear transport equations,wave equations,and Hamilton-Jacobi(HJ)equations.
查看更多>>摘要：Capturing elaborated flow structures and phenomena is required for well-solved numerical flows.The finite difference methods allow simple discretization of mesh and model equa-tions.However,they need simpler meshes,e.g.,rectangular.The inverse Lax-Wendroff(ILW)procedure can handle complex geometries for rectangular meshes.High-resolution and high-order methods can capture elaborated flow structures and phenomena.They also have strong mathematical and physical backgrounds,such as positivity-preserving,jump conditions,and wave propagation concepts.We perceive an effort toward direct numerical simulation,for instance,regarding weighted essentially non-oscillatory(WENO)schemes.Thus,we propose to solve a challenging engineering application without turbulence models.We aim to verify and validate recent high-resolution and high-order methods.To check the solver accuracy,we solved vortex and Couette flows.Then,we solved inviscid and viscous nozzle flows for a con-ical profile.We employed the finite difference method,positivity-preserving Lax-Friedrichs splitting,high-resolution viscous terms discretization,fifth-order multi-resolution WENO,ILW,and third-order strong stability preserving Runge-Kutta.We showed the solver is high-order and captured elaborated flow structures and phenomena.One can see oblique shocks in both nozzle flows.In the viscous flow,we also captured a free-shock separation,recircula-tion,entrainment region,Mach disk,and the diamond-shaped pattern of nozzle flows.
查看更多>>摘要：In this paper,we propose a novel Local Macroscopic Conservative(LoMaC)low rank ten-sor method with discontinuous Galerkin(DG)discretization for the physical and phase spaces for simulating the Vlasov-Poisson(VP)system.The LoMaC property refers to the exact local conservation of macroscopic mass,momentum,and energy at the discrete level.The recently developed LoMaC low rank tensor algorithm(arXiv:2207.00518)simultane-ously evolves the macroscopic conservation laws of mass,momentum,and energy using the kinetic flux vector splitting;then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables.This paper is a generalization of our previous work,but with DG discretization to take advan-tage of its compactness and flexibility in handling boundary conditions and its superior accuracy in the long term.The algorithm is developed in a similar fashion as that for a finite difference scheme,by observing that the DG method can be viewed equivalently in a nodal fashion.With the nodal DG method,assuming a tensorized computational grid,one will be able to(ⅰ)derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms,and(ⅱ)define a weighted inner product space based on the nodal DG grid points.The algorithm can be extended to the high dimensional problems by hierarchical Tucker(HT)decomposition of solution tensors and a correspond-ing conservative projection algorithm.In a similar spirit,the algorithm can be extended to DG methods on nodal points of an unstructured mesh,or to other types of discretization,e.g.,the spectral method in velocity direction.Extensive numerical results are performed to showcase the efficacy of the method.
查看更多>>摘要：The spread of an advantageous mutation through a population is of fundamental interest in population genetics.While the classical Moran model is formulated for a well-mixed popula-tion,it has long been recognized that in real-world applications,the population usually has an explicit spatial structure which can significantly influence the dynamics.In the context of can-cer initiation in epithelial tissue,several recent works have analyzed the dynamics of advanta-geous mutant spread on integer lattices,using the biased voter model from particle systems theory.In this spatial version of the Moran model,individuals first reproduce according to their fitness and then replace a neighboring individual.From a biological standpoint,the oppo-site dynamics,where individuals first die and are then replaced by a neighboring individual according to its fitness,are equally relevant.Here,we investigate this death-birth analogue of the biased voter model.We construct the process mathematically,derive the associated dual process,establish bounds on the survival probability of a single mutant,and prove that the pro-cess has an asymptotic shape.We also briefly discuss alternative birth-death and death-birth dynamics,depending on how the mutant fitness advantage affects the dynamics.We show that birth-death and death-birth formulations of the biased voter model are equivalent when fitness affects the former event of each update of the model,whereas the birth-death model is funda-mentally different from the death-birth model when fitness affects the latter event.
查看更多>>摘要：In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation laws.It can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary conditions.Furthermore,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite vol-ume HWENO reconstruction,and it can be regarded as a generalization of the one dimen-sional(1D)HWENO method.Extensive numerical tests are performed to verify the high resolution and high accuracy of the scheme.
查看更多>>摘要：Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accu-rate numerical schemes that can decouple these two equations.One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them.The resulting schemes are only first-order accurate in time,and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version.In this paper,we employ the spectral deferred correction method to improve the temporal accuracy,based on the first-order decoupled and energy-stable scheme con-structed by the stabilization idea.The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency.Within the framework of the spatially discretized local discontinuous Galerkin method,the resulting numerical schemes are fully decoupled,efficient,and high-order accurate in both time and space.Numerical experi-ments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.
查看更多>>摘要：This paper provides a study on the stability and time-step constraints of solving the lin-earized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discon-tinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Cou-rant-Friedrichs-Lewy(CFL)condition τ ≤(λ)h.Here,(λ)is the CFL number,τ is the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function of θ=d/h2 with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are pro-vided in the paper to illustrate the performance of IMEX methods under different time-step constraints.
查看更多>>摘要：The Blade Altering Toolbox(BAT)described in this paper is a tool designed for fast recon-struction of an altered blade geometry for design optimization purposes.The BAT algo-rithm is capable of twisting a given rotor's angle of attack and stretching the chord length along the span of the rotor.Several test cases were run using the BAT's algorithm.The BAT code's twisting,stretching,and mesh reconstruction capabilities proved to be able to handle reasonably large geometric alterations to a provided input rotor geometry.The test examples showed that the toolbox's algorithm could handle any stretching of the blade's chord as long as the blade remained within the original bounds of the unaltered mesh.The algorithm appears to fail when the net twist angle applied the geometry exceeds approxi-mately 30 degrees,however this limitation is dependent on the initial geometry and other input parameters.Overall,the algorithm is a very powerful tool for automating a design optimization procedure.
查看更多>>摘要：Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations.In this work,we analyze the stability properties of these methods and their sensitivity to the low-precision rounding errors,and demonstrate their performance in terms of accuracy and efficiency.We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed-precision additive Runge-Kutta(MP-ARK)methods.The convergence,accuracy,and runtime of these methods are explored.We show that for a given level of accuracy,suitably chosen MP-ARK methods may provide significant reductions in runtime.
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