Class of numerical methods for differential-algebraic systems with discontinuous right-hand sides

Numerical methods for Differential-Algebraic systems with discontinuous right-hand sides is discussed. A class of continuous Rosenbrock methods are constructed, and numerical experiments show that the continuous Rosenbrock methods are effective. Applying the methods, a fast and high-precision numerical algorithm is given to deal with typical discontinuous parts, which occur frequently in differential-algebraic systems(DAS).

ELLIPTIC EQUATIONS IN DIVERGENCE FORM WITH DISCONTINUOUS COEFFICIENTS IN DOMAINS WITH CORNERS

We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.

Particle Discontinuous Deformation Analysis of Static and Dynamic Crack Propagation in Brittle Material

Crack propagation in brittle material is not only crucial for structural safety evaluation,but also has a wideranging impact on material design,damage assessment,resource extraction,and scientific research.A thorough investigation into the behavior of crack propagation contributes to a better understanding and control of the properties of brittle materials,thereby enhancing the reliability and safety of both materials and structures.As an implicit discrete elementmethod,the Discontinuous Deformation Analysis(DDA)has gained significant attention for its developments and applications in recent years.Among these developments,the particle DDA equipped with the bonded particle model is a powerful tool for predicting the whole process of material from continuity to failure.The primary objective of this research is to develop and utilize the particle DDAtomodel and understand the complex behavior of cracks in brittle materials under both static and dynamic loadings.The particle DDA is applied to several classical crack propagation problems,including the crack branching,compact tensile test,Kalthoff impact experiment,and tensile test of a rectangular plate with a hole.The evolutions of cracks under various stress or geometrical conditions are carefully investigated.The simulated results are compared with the experiments and other numerical results.It is found that the crack propagation patterns,including crack branching and the formation of secondary cracks,can be well reproduced.The results show that the particle DDA is a qualified method for crack propagation problems,providing valuable insights into the fracture mechanism of brittle materials.

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

Application of modified discontinuous deformation analysis to dynamic modelling of the Baige landslide in the Jinsha River

Accurate dynamic modeling of landslides could help understand the movement mechanisms and guide disaster mitigation and prevention.Discontinuous deformation analysis(DDA)is an effective approach for investigating landslides.However,DDA fails to accurately capture the degradation in shear strength of rock joints commonly observed in high-speed landslides.In this study,DDA is modified by incorporating simplified joint shear strength degradation.Based on the modified DDA,the kinematics of the Baige landslide that occurred along the Jinsha River in China on 10 October 2018 are reproduced.The violent starting velocity of the landslide is considered explicitly.Three cases with different violent starting velocities are investigated to show their effect on the landslide movement process.Subsequently,the landslide movement process and the final accumulation characteristics are analyzed from multiple perspectives.The results show that the violent starting velocity affects the landslide motion characteristics,which is found to be about 4 m/s in the Baige landslide.The movement process of the Baige landslide involves four stages:initiation,high-speed sliding,impact-climbing,low-speed motion and accumulation.The accumulation states of sliding masses in different zones are different,which essentially corresponds to reality.The research results suggest that the modified DDA is applicable to similar high-level rock landslides.

Adaptive Sparse Grid Discontinuous Galerkin Method:Review and Software Implementation

This paper reviews the adaptive sparse grid discontinuous Galerkin(aSG-DG)method for computing high dimensional partial differential equations(PDEs)and its software implementation.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.

A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.

A Local Macroscopic Conservative(LoMaC)Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics

In this paper,we propose a novel Local Macroscopic Conservative(LoMaC)low rank tensor 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)simultaneously 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 advantage 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 corresponding 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.

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.

Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows

In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.

Superconvergence of Direct Discontinuous Galerkin Methods:Eigen-structure Analysis Based on Fourier Approach

This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.

HetNets Under Decoupled Uplink and Downlink Access with UE Random Discontinuous Transmission: Local Delay and Energy Efficiency

Due to the limited uplink capability in heterogeneousnetworks (HetNets), the decoupled uplinkand downlink access (DUDA) mode has recently beenproposed to improve the uplink performance. In thispaper, the random discontinuous transmission (DTX)at user equipment (UE) is adopted to reduce the interferencecorrelation across different time slots. By utilizingstochastic geometry, we analytically derive themean local delay and energy efficiency (EE) of an uplinkHetNet with UE random DTX scheme under theDUDA mode. These expressions are further approximatedas closed forms under reasonable assumptions.Our results reveal that under the DUDA mode, there isan optimal EE with respect to mute probability underthe finite local delay constraint. In addition, with thesame finite mean local delay as under the coupled uplinkand downlink access (CUDA) mode, the HetNetsunder the DUDA mode can achieve a higher EE witha lower mute probability.

High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models

In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.

Dynamical behaviors of recurrently connected neural networks and linearly coupled networks with discontinuous right-hand sides

The aim of this paper is to provide a sys- tematic review on the framework to analyze dynamics in recurrently connected neural networks with discontinu- ous right-hand sides with a focus on the authors＇ works in the past three years. The concept of the Filippov so- lution is employed to define the solution of the neural network systems by transforming them to differential in- clusions. The theory of viability provides a tool to study the existence and uniqueness of the solution and the Lya- punov function （functional） approach is used to investi- gate the global stability and synchronization. More pre- cisely, we prove that the diagonal-dominant conditions guarantee the existence, uniqueness, and stability of a general class of integro-differential equations with （al- most） periodic self-inhibitions, interconnection weights, inputs, and delays. This model is rather general and in- cludes the well-known Hopfield neural networks, Cohen- Grossberg neural networks, and cellular neural networks as special cases. We extend the absolute stability anal- ysis of gradient-like neural network model by relaxing the analytic constraints so that they can be employed to solve optimization problem with non-smooth cost func- tions. Furthermore, we study the global synchronization problem of a class of linearly coupled neural network with discontinuous right-hand sides.

Simulated and experimental investigation on discontinuous dynamic recrystallization of a near-α TA15 titanium alloy during isothermal hot compression in βsingle-phase field

A cellular automaton（CA） modeling of discontinuous dynamic recrystallization（DDRX） of a near-α Ti-6Al-2Zr-1Mo-1V（TA15） isothermally compressed in the β single phase field was presented.In the CA model,nucleation of the β-DDRX and the growth of recrystallized grains（re-grains） were considered and visibly simulated by the CA model.The driving force of re-grain growth was provided by dislocation density accumulating around the grain boundaries.To verify the CA model,the predicted flow stress by the CA model was compared with the experimental data.The comparison showed that the average relative errors were10.2%,10.1%and 6%,respectively,at 1.0,0.1 and 0.01 s^-1 of 1020 ℃,and were 10.2%,11.35%and 7.5%,respectively,at 1.0,0.1and 0.01 s^-1 of 1050 ℃.The CA model was further applied to predicting the average growth rate,average re-grain size and recrystallization kinetics.The simulated results showed that the average growth rate increases with the increasing strain rate or temperature,while the re-grain size increases with the decreasing strain rate;the volume fraction of recrystallization decreases with the increasing strain rate or decreasing temperature.

NUMERICAL SIMULATION OF UNSTEADY-STATE UNDEREXPANDED JET USING DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD

A discontinuous Galerkin finite element method （DG-FEM） is developed for solving the axisymmetric Euler equations based on two-dimensional conservation laws. The method is used to simulate the unsteady-state underexpanded axisymmetric jet. Several flow property distributions along the jet axis, including density, pres- sure and Mach number are obtained and the qualitative flowfield structures of interest are well captured using the proposed method, including shock waves, slipstreams, traveling vortex ring and multiple Mach disks. Two Mach disk locations agree well with computational and experimental measurement results. It indicates that the method is robust and efficient for solving the unsteady-state underexpanded axisymmetric jet.

Characterization of discontinuous yielding phenomenon in isothermal compression of TC8 titanium alloy

The discontinuous yielding phenomenon （DYP） during high temperature deformation was investigated based on the isothermal compression of TC8 titanium alloy. The DYP of TC8 titanium alloy was characterized by quantifying the yield drop of the DYP （△σUL） and ending strain of flow oscillation （εOSC） based on the flow stress？strain curves, and then the effect of deformation parameters on the △σUL and εOSC values was analyzed. The results show that the △σUL and εOSC values increase with the increase of strain rate. The effect of deformation temperature on the ？σUL value depends on the strain rate. Finally, the transmission electron microscope （TEM） observation shows the evidence for the dynamic theory, which ascribes the DYP to the generation of mobile dislocation at the grain boundary. Meanwhile, the optical microscope （OM） observation shows that both the primary α grain and β grain become smaller with the increase of strain, which well interprets the effect of deformation parameters on the △σUL and εOSC values.

NUMERICAL INVESTIGATION OF TOROIDAL SHOCK WAVES FOCUSING USING DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD

A numerical simulation of the toroidal shock wave focusing in a co-axial cylindrical shock tube is inves- tigated by using discontinuous Galerkin （DG） finite element method to solve the axisymmetric Euler equations. For validating the numerical method, the shock-tube problem with exact solution is computed, and the computed results agree well with the exact cases. Then, several cases with higher incident Mach numbers varying from 2.0 to 5.0 are simulated. Simulation results show that complicated flow-field structures of toroidal shock wave diffraction, reflection, and focusing in a co-axial cylindrical shock tube can be obtained at different incident Mach numbers and the numerical solutions appear steep gradients near the focusing point, which illustrates the DG method has higher accuracy and better resolution near the discontinuous point. Moreover, the focusing peak pres- sure with different grid scales is compared.

ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.

FORMULATIONS OF THE THREE-DIMENSIONAL DISCONTINUOUS DEFORMATION ANALYSIS METHOD

This paper extends the original 2D discontinuous deformation analysis(DDA)method proposed by Shi to 3D cases,and presents the formulations of the 3D DDA.The formulations maintain the characteristics of the original 2D DDA approach.Contacts between the blocks are detected by using Common-Plane (C-P) approach and the non-smooth contact,such as of vertex-to-vertex,vertex- to-edge and edge-to-edge types,can be handled easily based on the C-P method.The matrices of equilibrium equations have been given in detail for programming purposes.The C program codes for the 3D DDA are developed.The ability and accuracy of the formulations and the program are verified by the analytical solutions of several dynamic examples.The robustness and versatility of the algorithms presented in this paper are demonstrated with the aid of an example of scattering of densely packed cubes.Finally,implications and future extensions are discussed.