Adaptive Mesh Refinement for conservative systems: multi-dimensional efficiency evaluation

Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro- and magnetohydrodynamic simulations, AMR is used in combination with several shock-capturing, conservative discretization schemes. Solution accuracy and execution times are compared with static grid simulations at the corresponding high resolution and time spent on AMR overhead is reported. Our examples reach corresponding efficiencies of 5 to 20 in multi-dimensional calculations and only 1.5-8% overhead is observed. For AMR calculations of multi-dimensional magnetohydrodynamic problems, several strategies for controlling the constraint are examined. Three source term approaches suitable for cell-centered representations are shown to be effective. For 2D and 3D calculations where a transition to a more globally turbulent state takes place, it is advocated to use an approximate Riemann solver based discretization at the highest allowed level(s), in combination with the robust Total Variation Diminishing Lax-Friedrichs method on the coarser levels. This level-dependent use of the spatial discretization acts as a computationally efficient, hybrid scheme.     

Simulation of the Paraxial Laser Propagation Coupled with Hydrodynamics in 3D Geometry

We address here numerical simulation problems for modeling some phenomena arising in plasmas produced in experimental devices for Inertial Confinement Fusion. The model consists of a compressible fluid dynamics system coupled with a paraxial equation for modeling the laser propagation. For the fluid dynamics system, a numerical method of Lagrange–Euler type is used. For the paraxial equation, a time implicit discretization is settled which preserves the laser energy balance; the method is based on a splitting of the propagation term and the diffraction terms according to the propagation spatial variable. We give some features on the 3D implementation of the method in the parallel platform HERA. Results showing the accuracy of the numerical scheme are presented and we give also numerical results related to cases corresponding to realistic simulations, with a mesh containing up to 500 millions of cells.     

High Accuracy Compact Schemes and Gibbs' Phenomenon

Compact difference schemes have been investigated for their ability to capture discontinuities. A new proposed scheme (Sengupta, Ganerwal and De (2003). J. Comp. Phys. 192(2), 677.) is compared with another from the literature Zhong (1998). J. Comp. Phys. 144, 622 that was developed for hypersonic transitional flows for their property related to spectral resolution and numerical stability. Solution of the linear convection equation is obtained that requires capturing discontinuities. We have also studied the performance of the new scheme in capturing discontinuous solution for the Burgers equation. A very simple but an effective method is proposed here in early diagnosis for evanescent discontinuities. At the discontinuity, we switch to a third order one-sided stencil, thereby retaining the high accuracy of solution. This produces solution with vastly reduced Gibbs' phenomenon of the solution. The essential causes behind Gibbs' phenomenon is also explained.     

Hybrid Finite Difference Weighted Essentially Non-oscillatory Schemes for the Compressible Ideal Magnetohydrodynamics Equation

In this paper, we present hybrid weighted essentially non-oscillatory (WENO) schemes with several discontinuity detectors for solving the compressible ideal magnetohydrodynamics (MHD) equation. Li and Qiu (J Comput Phys 229:8105–8129, 2010) examined effectiveness and efficiency of several different troubled-cell indicators in hybrid WENO methods for Euler gasdynamics. Later, Li et al. (J Sci Comput 51:527–559, 2012) extended the hybrid methods for solving the shallow water equations with four better indicators. Hybrid WENO schemes reduce the computational costs, maintain non-oscillatory properties and keep sharp transitions for problems. The numerical results of hybrid WENO-JS/WENO-M schemes are presented to compare the ability of several troubled-cell indicators with a variety of test problems. The focus of this paper, we propose optimal and reliable indicators for performance comparison of hybrid method using troubled-cell indicators for efficient numerical method of ideal MHD equations. We propose a modified ATV indicator that uses a second derivative. It is advantageous for differential discontinuity detection such as jump discontinuity and kink. A detailed numerical study of one-dimensional and two-dimensional cases is conducted to address efficiency (CPU time reduction and more accurate numerical solution) and non-oscillatory property problems. We demonstrate that the hybrid WENO-M scheme preserves the advantages of WENO-M and the ratio of computational costs of hybrid WENO-M and hybrid WENO-JS is smaller than that of WENO-M and WENO-JS.     

A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems

A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are \(L^2\) stable even without interior penalty. For time discretization, we use Crank–Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal \(L^2\) error estimate of \(O(h^{k+1})\) for polynomials of degree k for semi-discrete DG schemes, and the \(L^2\) error of \(O(h^{k+1} +(\Delta t)^2)\) for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift–Hohenberg equation endowed with a decay free energy is presented.     

Implementation of an Algorithm for Finding Analyticity Conditions for a Solution of a Difference Equation

A linear difference operator L with polynomial coefficients and a function F(x) satisfying the equation LF(x) = 0 are considered. The function is assumed to be analytic in the interval (–, d), where > 0. In the paper, an implementation of an algorithm suggested by S.A. Abramov and M. van Hoeij for finding conditions that guarantee analyticity of F(x) on the entire real axis is presented. The analyticity conditions are linear relations for values of F(x) and its derivatives at a given point belonging to the half-interval [0, d). A procedure for computing values of F(x) and its derivatives up to a prescribed order at a given point x ₀ is also implemented. Examples illustrating the program operation are presented.     

Optimal vorticity accuracy in an efficient velocity–vorticity method for the 2D Navier–Stokes equations

We study a velocity–vorticity scheme for the 2D incompressible Navier–Stokes equations, which is based on a formulation that couples the rotation form of the momentum equation with the vorticity equation, and a temporal discretization that stably decouples the system at each time step and allows for simultaneous solving of the vorticity equation and velocity–pressure system (thus if special care is taken in its implementation, the method can have no extra cost compared to common velocity–pressure schemes). This scheme was recently shown to be unconditionally long-time \(H^1\) stable for both velocity and vorticity, which is a property not shared by any common velocity–pressure method. Herein, we analyze the scheme’s convergence, and prove that it yields unconditional optimal accuracy for both velocity and vorticity, thus making it advantageous over common velocity–pressure schemes if the vorticity variable is of interest. Numerical experiments are given that illustrate the theory and demonstrate the scheme’s usefulness on some benchmark problems.     

On the unification power of models

In November 2000, the OMG made public the MDAinitiative, a particular variant of a new global trend called MDE (Model Driven Engineering). The basic ideas of MDA are germane to many other approaches such as generative programming, domain specific languages, model-integrated computing, generic model management, software factories, etc. MDA may be defined as the realization of MDE principles around a set of OMG standards like MOF, XMI, OCL, UML, CWM, SPEM, etc. MDE is presently making several promises about the potential benefits that could be reaped from a move from code-centric to model-based practices. When we observe these claims, we may wonder when they may be satisfied: on the short, medium or long term or even never perhaps for some of them. This paper tries to propose a vision of the development of MDE based on some lessons learnt in the past 30 years in the development of object technology. The main message is that a basic principle (Everything is an object) was most helpful in driving the technology in the direction of simplicity, generality and power of integration. Similarly in MDE, the basic principle that Everything is a model has many interesting properties, among others the capacity to generate a realistic research agenda. We postulate here that two core relations (representation and conformance) are associated to this principle, as inheritance and instantiation were associated to the object unification principle in the class-based languages of the 80s. We suggest that this may be most useful in understanding many questions about MDE in general and the MDA approach in particular. We provide some illustrative examples. The personal position taken in this paper would be useful if it could generate a critical debate on the research directions in MDE.     

Implications of the Dirichlet Assumption for Discretization of Continuous Variables in Naive Bayesian Classifiers

In a naive Bayesian classifier, discrete variables as well as discretized continuous variables are assumed to have Dirichlet priors. This paper describes the implications and applications of this model selection choice. We start by reviewing key properties of Dirichlet distributions. Among these properties, the most important one is perfect aggregation, which allows us to explain why discretization works for a naive Bayesian classifier. Since perfect aggregation holds for Dirichlets, we can explain that in general, discretization can outperform parameter estimation assuming a normal distribution. In addition, we can explain why a wide variety of well-known discretization methods, such as entropy-based, ten-bin, and bin-log l, can perform well with insignificant difference. We designed experiments to verify our explanation using synthesized and real data sets and showed that in addition to well-known methods, a wide variety of discretization methods all perform similarly. Our analysis leads to a lazy discretization method, which discretizes continuous variables according to test data. The Dirichlet assumption implies that lazy methods can perform as well as eager discretization methods. We empirically confirmed this implication and extended the lazy method to classify set-valued and multi-interval data with a naive Bayesian classifier.     

A Fast Direct Solver for a Class of Elliptic Partial Differential Equations

We describe a fast and robust method for solving the large sparse linear systems that arise upon the discretization of elliptic partial differential equations such as Laplace’s equation and the Helmholtz equation at low frequencies. While most existing fast schemes for this task rely on so called “iterative” solvers, the method described here solves the linear system directly (to within an arbitrary predefined accuracy). The method is described for the particular case of an operator defined on a square uniform grid, but can be generalized other geometries. For a grid containing N points, a single solve requires O(Nlog ² N) arithmetic operations and storage. Storing the information required to perform additional solves rapidly requires O(Nlog N) storage. The scheme is particularly efficient in situations involving domains that are loaded on the boundary only and where the solution is sought only on the boundary. In this environment, subsequent solves (after the first) can be performed in operations. The efficiency of the scheme is illustrated with numerical examples. For instance, a system of size 10⁶×10⁶ is directly solved to seven digits accuracy in four minutes on a 2.8 GHz P4 desktop PC.     

Unconditional $$L^{\infty }$$-convergence of two compact conservative finite difference schemes for the nonlinear Schrödinger equation in multi-dimensions

This paper is concerned with the unconditional and optimal \(L^{\infty }\)-error estimates of two fourth-order (in space) compact conservative finite difference time domain schemes for solving the nonlinear Schrödinger equation in two or three space dimensions. The fact of high space dimension and the approximation via compact finite difference discretization bring difficulties in the convergence analysis. The two proposed schemes preserve the total mass and energy in the discrete sense. To establish the optimal convergence results without any constraint on the time step, besides the standard energy method, the cut-off function technique as well as a ‘lifting’ technique are introduced. On the contrast, previous works in the literature often require certain restriction on the time step. The convergence rate of the proposed schemes are proved to be of \(O(h^4+\tau ^2)\) with time step \(\tau \) and mesh size h in the discrete \(L^{\infty }\)-norm. The analysis method can be directly extended to other finite difference schemes for solving the nonlinear Schrödinger-type equations. Numerical results are reported to support our theoretical analysis, and investigate the effect of the nonlinear term and initial data on the blow-up solution.     

All Mach Number Second Order Semi-implicit Scheme for the Euler Equations of Gas Dynamics

This paper presents an asymptotic preserving (AP) all Mach number finite volume shock capturing method for the numerical solution of compressible Euler equations of gas dynamics. Both isentropic and full Euler equations are considered. The equations are discretized on a staggered grid. This simplifies flux computation and guarantees a natural central discretization in the low Mach limit, thus dramatically reducing the excessive numerical diffusion of upwind discretizations. Furthermore, second order accuracy in space is automatically guaranteed. For the time discretization we adopt an Semi-IMplicit/EXplicit (S-IMEX) discretization getting an elliptic equation for the pressure in the isentropic case and for the energy in the full Euler case. Such equations can be solved linearly so that we do not need any iterative solver thus reducing computational cost. Second order in time is obtained by a suitable S-IMEX strategy taken from Boscarino et al. (J Sci Comput 68:975–1001, 2016). Moreover, the CFL stability condition is independent of the Mach number and depends essentially on the fluid velocity. Numerical tests are displayed in one and two dimensions to demonstrate performance of our scheme in both compressible and incompressible regimes.     

A Level Set Algorithm for Tracking Discontinuities in Hyperbolic Conservation Laws II: Systems of Equations

A level set algorithm for tracking discontinuities in hyperbolic conservation laws is presented. The algorithm uses a simple finite difference approach, analogous to the method of lines scheme presented in [36]. The zero of a level set function is used to specify the location of the discontinuity. Since a level set function is used to describe the front location, no extra data structures are needed to keep track of the location of the discontinuity. Also, two solution states are used at all computational nodes, one corresponding to the real state, and one corresponding to a ghost node state, analogous to the Ghost Fluid Method of [12]. High order pointwise convergence was demonstrated for scalar linear and nonlinear conservation laws, even at discontinuities and in multiple dimensions in the first paper of this series [3]. The solutions here are compared to standard high order shock capturing schemes, when appropriate. This paper focuses on the issues involved in tracking discontinuities in systems of conservation laws. Examples will be presented of tracking contacts and hydrodynamic shocks in inert and chemically reacting compressible flow.     

Stability Analysis and Error Estimates of Semi-implicit Spectral Deferred Correction Coupled with Local Discontinuous Galerkin Method for Linear Convection–Diffusion Equations

In this paper, we focus on the theoretical analysis of the second and third order semi-implicit spectral deferred correction (SDC) time discretization with local discontinuous Galerkin (LDG) spatial discretization for the one-dimensional linear convection–diffusion equations. We mainly study the stability and error estimates of the corresponding fully discrete scheme. Based on the Picard integral equation, the SDC method is driven iteratively by either the explicit Euler method or the implicit Euler method. It is easy to implement for arbitrary order of accuracy. For the semi-implicit SDC scheme, the iteration and the left-most endpoint involved in the integral for the implicit part increase the difficulty of the theoretical analysis. To be more precise, the test functions are more complex and the energy equations are more difficult to construct, compared with the Runge–Kutta type semi-implicit schemes. Applying the energy techniques, we obtain both the second and third order semi-implicit SDC time discretization with LDG spatial discretization are stable provided the time step \(\tau \le \tau _{0}\), where the positive \(\tau _{0}\) depends on the diffusion and convection coefficients and is independent of the mesh size h. We then obtain the optimal error estimates for the corresponding fully discrete scheme under the condition \(\tau \le \tau _{0}\) with similar technique for stability analysis. Numerical examples are presented to illustrate our theoretical results.     

A Study of Reinforcement Learning in the Continuous Case by the Means of Viscosity Solutions

This paper proposes a study of Reinforcement Learning (RL) for continuous state-space and time control problems, based on the theoretical framework of viscosity solutions (VSs). We use the method of dynamic programming (DP) which introduces the value function (VF), expectation of the best future cumulative reinforcement. In the continuous case, the value function satisfies a non-linear first (or second) order (depending on the deterministic or stochastic aspect of the process) differential equation called the Hamilton-Jacobi-Bellman (HJB) equation. It is well known that there exists an infinity of generalized solutions (differentiable almost everywhere) to this equation, other than the VF. We show that gradient-descent methods may converge to one of these generalized solutions, thus failing to find the optimal control.In order to solve the HJB equation, we use the powerful framework of viscosity solutions and state that there exists a unique viscosity solution to the HJB equation, which is the value function. Then, we use another main result of VSs (their stability when passing to the limit) to prove the convergence of numerical approximations schemes based on finite difference (FD) and finite element (FE) methods. These methods discretize, at some resolution, the HJB equation into a DP equation of a Markov Decision Process (MDP), which can be solved by DP methods (thanks to a strong contraction property) if all the initial data (the state dynamics and the reinforcement function) were perfectly known. However, in the RL approach, as we consider a system in interaction with some a priori (at least partially) unknown environment, which learns from experience, the initial data are not perfectly known but have to be approximated during learning. The main contribution of this work is to derive a general convergence theorem for RL algorithms when one uses only approximations (in a sense of satisfying some weak contraction property) of the initial data. This result can be used for model-based or model-free RL algorithms, with off-line or on-line updating methods, for deterministic or stochastic state dynamics (though this latter case is not described here), and based on FE or FD discretization methods. It is illustrated with several RL algorithms and one numerical simulation for the Car on the Hill problem.     

Non-Linear Filtering and Limiting in High Order Methods for Ideal and Non-Ideal MHD

The adaptive nonlinear filtering and limiting in spatially high order schemes (Yee et al. J. Comput. Phys. 150, 199–238, (1999), Sjögreen and Yee, J. Scient. Comput. 20, 211–255, (2004)) for the compressible Euler and Navier–Stokes equations have been recently extended to the ideal and non-ideal magnetohydrodynamics (MHD) equations, (Sjögreen and Yee, (2003), Proceedings of the 16th AIAA/CFD conference, June 23–26, Orlando F1; Yee and Sjögreen (2003), Proceedings of the International Conference on High Performance Scientific Computing, March, 10–14, Honai, Vietnam; Yee and Sjögreen (2003), RIACS Technical Report TR03. 10, July, NASA Ames Research Center; Yee and Sjögreen (2004), Proceedings of the ICCF03, July 12–16, Toronto, Canada). The numerical dissipation control in these adaptive filter schemes consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free from numerical dissipation contamination. The numerical dissipation considered consists of high order linear dissipation for the suppression of high frequency oscillation and the nonlinear dissipative portion of high-resolution shock-capturing methods for discontinuity capturing. The applicable nonlinear dissipative portion of high-resolution shock-capturing methods is very general. The objective of this paper is to investigate the performance of three commonly used types of discontinuity capturing nonlinear numerical dissipation for both the ideal and non-ideal MHD.     

Unconditionally Optimal Error Estimates of a Linearized Galerkin Method for Nonlinear Time Fractional Reaction–Subdiffusion Equations

This paper is concerned with unconditionally optimal error estimates of linearized Galerkin finite element methods to numerically solve some multi-dimensional fractional reaction–subdiffusion equations, while the classical analysis for numerical approximation of multi-dimensional nonlinear parabolic problems usually require a restriction on the time-step, which is dependent on the spatial grid size. To obtain the unconditionally optimal error estimates, the key point is to obtain the boundedness of numerical solutions in the \(L^\infty \)-norm. For this, we introduce a time-discrete elliptic equation, construct an energy function for the nonlocal problem, and handle the error summation properly. Compared with integer-order nonlinear problems, the nonlocal convolution in the time fractional derivative causes much difficulties in developing and analyzing numerical schemes. Numerical examples are given to validate our theoretical results.     

A hierarchy theorem for multihead stack-counter automata

Summary Let L _b = {w ₁ ^*...* w _2b ¦w _i is in {0,1}^* and w _i = w _2b+1–i for 1i2b for b1. We show that the language L _b is not recognizable by any nondeterministic one-way k-head stack-counter automata if \left( {\begin{array}{*{20}c} k \\ 2 \\ \end{array} } \right)$$ " align="middle" border="0"> . As a corollary, we show that k+1 heads are better than k for one-way multihead stack-counter automata.     

Multi-symplectic method for peakon–antipeakon collision of quasi-Degasperis–Procesi equation

Focusing on the local geometric properties of the shockpeakon for the Degasperis–Procesi equation, a multi-symplectic method for the quasi-Degasperis–Procesi equation is proposed to reveal the jump discontinuity of the shockpeakon for the Degasperis–Procesi equation numerically in this paper. The main contribution of this paper lies in the following: (1) the uniform multi-symplectic structure of the b-family equation is constructed; (2) the stable jump discontinuity of the shockpeakon for the Degasperis–Procesi equation is reproduced by simulating the peakon–antipeakon collision process of the quasi-Degasperis–Procesi equation. First, the multi-symplectic structure and several local conservation laws are presented for the b-family equation with two exceptions (b=3$b=3$ and b=4$b=4$). And then, the Preissman Box multi-symplectic scheme for the multi-symplectic structure is constructed and the mathematical proofs for the discrete local conservation laws of the multi-symplectic structure are given. Finally, the numerical experiments on the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation are reported to investigate the jump discontinuity of shockpeakon of the Degasperis–Procesi equation. From the numerical results, it can be concluded that the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation can be simulated well by the multi-symplectic method and the simulation results can reveal the jump discontinuity of shockpeakon of the Degasperis–Procesi equation approximately.