Weights Design For Maximal Order WENO Schemes

Weighted essentially non-oscillatory (WENO) finite difference schemes, developed by Liu et al. (Comput Phys 115(1):200–212, 1994) and improved by Jiang and Shu (Comput Phys 126(1):202–228, 1996), are one of the most popular methods to approximate the solutions of hyperbolic equations. But these schemes fail to provide maximal order accuracy near smooth extrema, where the first derivative of the solution becomes zero. Some authors have addressed this problem with different weight designs. In this paper we focus on the weights proposed by Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009). They propose new weights to provide faster weight convergence than those presented in Borges et al. (J Comput Phys 227:3191–3211, 2008) and deduce some constraints on the weights parameters to guarantee that the WENO scheme has maximal order for sufficiently smooth solutions with an arbitrary number of vanishing derivatives. We analyze the scheme with the weights proposed in Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009) and prove that near discontinuities it achieves worse orders than classical WENO schemes. In order to solve these accuracy problems, we define new weights, based on those proposed in Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009), and get some constraints on the weights parameters to guarantee maximal order accuracy for the resulting schemes.     

Arbitrary High-Order Explicit Hybridizable Discontinuous Galerkin Methods for the Acoustic Wave Equation

We propose a new formulation of explicit time integration for the hybridizable discontinuous Galerkin (HDG) method in the context of the acoustic wave equation based on the arbitrary derivative approach. The method is of arbitrary high order in space and time without restrictions such as the Butcher barrier for Runge–Kutta methods. To maintain the superconvergence property characteristic for HDG spatial discretizations, a special reconstruction step is developed, which is complemented by an adjoint consistency analysis. For a given time step size, this new method is twice as fast compared to a low-storage Runge–Kutta scheme of order four with five stages at polynomial degrees between two and four. Several numerical examples are performed to demonstrate the convergence properties, reveal dispersion and dissipation errors, and show solution behavior in the presence of material discontinuities. Also, we present the combination of local time stepping with h-adaptivity on three-dimensional meshes with curved elements.     

Strong Stability for Runge–Kutta Schemes on a Class of Nonlinear Problems

In this paper we consider Strong Stability Preserving (SSP) properties for explicit Runge–Kutta (RK) methods applied to a class of nonlinear ordinary differential equations. We define new modified threshold factors that allow us to prove properties, provided that they hold for explicit Euler steps. For many methods, the stepsize restrictions obtained are sharper than the ones obtained in terms of the Kraaijevanger’s coefficient in the SSP theory. In particular, for the classical 4-stage fourth order method we get nontrivial stepsize restrictions. Furthermore, the order barrier $p\le 4$ for explicit SSP RK methods is not obtained. An open question is the existence of explicit RK schemes with order $p\ge 5$ and nontrivial modified threshold factor. The numerical experiments done illustrate the results obtained.     

A Maximum-Principle-Satisfying High-Order Finite Volume Compact WENO Scheme for Scalar Conservation Laws with Applications in Incompressible Flows

In this paper, a maximum-principle-satisfying finite volume compact scheme is proposed for solving scalar hyperbolic conservation laws. The scheme combines weighted essentially non-oscillatory schemes (WENO) with a class of compact schemes under a finite volume framework, in which the nonlinear WENO weights are coupled with lower order compact stencils. The maximum-principle-satisfying polynomial rescaling limiter in Zhang and Shu (J Comput Phys 229:3091–3120, 2010, Proc R Soc A Math Phys Eng Sci 467:2752–2776, 2011) is adopted to construct the present schemes at each stage of an explicit Runge–Kutta method, without destroying high order accuracy and conservativity. Numerical examples for one and two dimensional problems including incompressible flows are presented to assess the good performance, maximum principle preserving, essentially non-oscillatory and high resolution of the proposed method.     

Metodi ad un passo fortemente stabili per equazioni integrali di Volterra di seconda specie di tipo stiff

In this paper we propose some implicit methods for stiff Volterra integral equations of second kind. The methods are constructed on the integro-differential equation obtained by differentiation of the Volterra equation. The numerical schemes are derived using a class of A-stable and L-stable methods for ordinary differential equations (proposed by Liniger and Willoughby in [3]) associated with the Gregory quadrature formula. Related to the test equation: $$y(t) = 1 + \int_0^t {[\lambda + \mu (t - s)] y(s)ds \lambda , \mu< 0} $$ we give the definition ofA-stability and ?-stability for the proposed numerical methods how natural extension of the A-stability and L-stability for the schemes for solving ordinary differential equations. We show how we have to choose the parameters of the methods in order to obtainA-stability and ?-stability schemes. Because of these properties the proposed schemes are particularly advantageous in the case of stiff Volterra integral equations.     

A note on the approximation of mean-payoff games

We consider the problem of designing approximation schemes for the values of mean-payoff games. It was recently shown that (1) mean-payoff with rational weights scaled on [−1,1]$[-1,1]$ admit additive fully-polynomial approximation schemes, and (2) mean-payoff games with positive weights admit relative fully-polynomial approximation schemes. We show that the problem of designing additive/relative approximation schemes for general mean-payoff games (i.e. with no constraint on their edge-weights) is P-time equivalent to determining their exact solution.     

Accurate Implicit–Explicit General Linear Methods with Inherent Runge–Kutta Stability

We investigate implicit–explicit (IMEX) general linear methods (GLMs) with inherent Runge–Kutta stability (IRKS) for differential systems with non-stiff and stiff processes. The construction of such formulas starts with implicit GLMs with IRKS which are A- and L-stable, and then we ‘remove’ implicitness in non-stiff terms by extrapolating unknown stage derivatives by stage derivatives which are already computed by the method. Then we search for IMEX schemes with large regions of absolute stability of the ‘explicit part’ of the method assuming that the ‘implicit part’ of the scheme is \(A(\alpha )\)-stable for some \(\alpha \in (0,\pi /2]\). Examples of highly stable IMEX GLMs are provided of order \(1\le p\le 4\). Numerical examples are also given which illustrate good performance of these schemes.     

An Approximate Solution for a Class of Second-Order Elliptic Variational Inequalities in Arbitrary-Form Domains

Penalty and dummy-domain methods are used to approximate second-order elliptic variational inequalities with a restriction inside a domain by nonlinear boundary-value problems in a rectangle. Difference schemes, with the order of accuracy O(h ^1/2) in the grid norm W ₂ ¹(), are constructed for these problems.     

On a Class of Implicit–Explicit Runge–Kutta Schemes for Stiff Kinetic Equations Preserving the Navier–Stokes Limit

Implicit–explicit (IMEX) Runge–Kutta (RK) schemes are popular high order time discretization methods for solving stiff kinetic equations. As opposed to the compressible Euler limit (leading order asymptotics of the Boltzmann equation as the Knudsen number \(\varepsilon \) goes to zero), their asymptotic behavior at the Navier–Stokes (NS) level (next order asymptotics) was rarely studied. In this paper, we analyze a class of existing IMEX RK schemes and show that, under suitable initial conditions, they can capture the NS limit without resolving the small parameter \(\varepsilon \), i.e., \(\varepsilon =o(\Delta t)\), \(\Delta t^m=o(\varepsilon )\), where m is the order of the explicit RK part in the IMEX scheme. Extensive numerical tests for BGK and ES-BGK models are performed to verify our theoretical results.     

Fully discrete potential-based finite element methods for a transient eddy current problem

Fully discrete potential-based finite element methods called methods are used to solve a transient eddy current problem in a three-dimensional convex bounded polyhedron. Using methods, fully discrete coupled and decoupled numerical schemes are developed. The existence and uniqueness of solutions for these schemes together with the energy-norm error estimates are provided. To verify the validity of both schemes, some computer simulations are performed for the model from TEAM Workshop Problem 7. This work was supported by Postech BSRI Research Fund-2009, National Basic Research Program of China (2008CB425701), NSFC under the grant 10671025 and the Key Project of Chinese Ministry of Education (No. 107018).     

A P_N P_M{-} CPR Framework for Hyperbolic Conservation Laws

The correction procedure via reconstruction (CPR) method is a discontinuous nodal formulation unifying several well-known methods in a simple finite difference like manner. The \(P_NP_M{-} CPR \) formulation is an extension of \(P_NP_M\) or the reconstructed discontinuous Galerkin (RDG) method to the CPR framework. It is a hybrid finite volume and discontinuous Galerkin (DG) method, in which neighboring cells are used to build a higher order polynomial than the solution representation in the cell under consideration. In this paper, we present several \(P_NP_M\) schemes under the CPR framework. Many interesting schemes with various orders of accuracy and efficiency are developed. The dispersion and dissipation properties of those methods are investigated through a Fourier analysis, which shows that the \(P_NP_M{-} CPR \) method is dependent on the position of the solution points. Optimal solution points for 1D \(P_NP_M{-} CPR \) schemes which can produce expected order of accuracy are identified. In addition, the \(P_NP_M{-} CPR \) method is extended to solve 2D inviscid flow governed by the Euler equations and several numerical tests are performed to assess its performance.     

Neural-Network-Based State Feedback Control of a Nonlinear Discrete-Time System in Nonstrict Feedback Form

In this paper, a suite of adaptive neural network (NN) controllers is designed to deliver a desired tracking performance for the control of an unknown, second-order, nonlinear discrete-time system expressed in nonstrict feedback form. In the first approach, two feedforward NNs are employed in the controller with tracking error as the feedback variable whereas in the adaptive critic NN architecture, three feedforward NNs are used. In the adaptive critic architecture, two action NNs produce virtual and actual control inputs, respectively, whereas the third critic NN approximates certain strategic utility function and its output is employed for tuning action NN weights in order to attain the near-optimal control action. Both the NN control methods present a well-defined controller design and the noncausal problem in discrete-time backstepping design is avoided via NN approximation. A comparison between the controller methodologies is highlighted. The stability analysis of the closed-loop control schemes is demonstrated. The NN controller schemes do not require an offline learning phase and the NN weights can be initialized at zero or random. Results show that the performance of the proposed controller schemes is highly satisfactory while meeting the closed-loop stability.      

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.     

On-line Prediction and Conversion Strategies

We study the problem of deterministically predicting boolean values by combining the boolean predictions of several experts. Previous on-line algorithms for this problem predict with the weighted majority of the experts' predictions. These algorithms give each expert an exponential weight ^m where is a constant in [0, 1) andm is the number of mistakes made by the expert in the past. We show that it is better to use sums of binomials as weights. In particular, we present a deterministic algorithm using binomial weights that has a better worst case mistake bound than the best deterministic algorithm using exponential weights. The binomial weights naturally arise from a version space argument. We also show how both exponential and binomial weighting schemes can be used to make prediction algorithms robust against noise.     

Stability and Convergence of Modified Du Fort–Frankel Schemes for Solving Time-Fractional Subdiffusion Equations

A class of modified Du Fort–Frankel-type schemes is investigated for fractional subdiffusion equations in the Jumarie’s modified Riemann–Liouville form with constant, variable or distributed fractional order. New explicit difference methods are constructed by combining the \(L1\) approximation of the modified fractional derivative with the idea of Du Fort–Frankel scheme, well-known for ordinary diffusion equations. Unconditional stability of the explicit methods is established in the sense of a discrete energy norm. The proposed schemes are shown to be convergent under the time-step (consistency) restriction of the classical Du Fort–Frankel scheme. Numerical examples are included to support our theoretical results.     

A modified fifth-order WENO scheme for hyperbolic conservation laws

Recently a WENO scheme, with smoothness indicators constructed based on $L_1$ measure is introduced by Ha et al. (2013) and the improved version of this scheme is presented by Kim et al. (2016), referred to as WENO-NS and WENO-P schemes respectively. These schemes perform better than the existing many fifth-order WENO schemes for the problems which contain discontinuities and attain fifth-order accuracy at the critical points where the first derivative vanishes but not at the points where the second derivatives are zero. This paper deals with modification of the above said methods to obtain a new fifth-order weighted essentially non-oscillatory (WENO) scheme. A new global-smoothness indicator is proposed which shows an improved behavior over the solutions of WENO-NS and WENO-P schemes and the proposed scheme attains an optimal order of approximation, even at the critical points where the first and second derivatives vanish but not the third derivative. Examples are taken in the numeric section to check the robustness and accuracy of the proposed scheme for one and two-dimensional system of Euler equations.