AENO: a Novel Reconstruction Method in Conjunction with ADER Schemes for Hyperbolic Equations

In this paper, we present a novel spatial reconstruction scheme, called AENO, that results from a special averaging of the ENO polynomial and its closest neighbour, while retaining the stencil direction decided by the ENO choice. A variant of the scheme, called m-AENO, results from averaging the modified ENO (m-ENO) polynomial and its closest neighbour. The concept is thoroughly assessed for the one-dimensional linear advection equation and for a one-dimensional non-linear hyperbolic system, in conjunction with the fully discrete, high-order ADER approach implemented up to fifth order of accuracy in both space and time. The results, as compared to the conventional ENO, m-ENO and WENO schemes, are very encouraging. Surprisingly, our results show that the $$L_{1}$$ L 1 -errors of the novel AENO approach are the smallest for most cases considered. Crucially, for a chosen error size, AENO turns out to be the most efficient method of all five methods tested.

CABARET scheme with improved dispersion properties for systems of linear hyperbolic-type differential equations

Предложен балансно-характеристический метод решения систем линейных дифференциальных уравнений в частных производных гиперболического типа, обладающий четвертым порядком аппроксимации на равномерных сетках и вторым порядком и улучшенными дисперсионными свойствами на неравномерных сетках. Метод основан на известной схеме КАБАРЕ, балансные фазы которой модифицированы путем добавления антидисперсионных членов особого вида. Ранее метод, обладающий схожими свойствами, предлагался только для простейшего одномерного линейного уравнения переноса. Приведенная модификация схемы позволяет улучшить дисперсионные свойства переноса сразу всех инвариантов Римана рассматриваемой системы уравнений. Схема бездиссипативна при отключенных процедурах монотонизации и устойчива при числах Куранта CFL ≤ 1. Точность метода и его порядок сходимости продемонстрированы на серии расчетов задачи о переносе волны, промодулированной гауссианом, на последовательности сгущающихся сеток. Предложенный метод планируется использовать в качестве основы для построения схемы КАБАРЕ с улучшенными дисперсионными свойствами для систем нелинейных дифференциальных уравнений. A conservative-characteristic method to solve systems of linear hyperbolic-type partial differential equations is proposed. This method has the fourth order of approximation on uniform grids and the second approximation order and improved dispersion properties on non-uniform grids. The proposed method is based on the well-known CABARET scheme whose conservative phases are modified by adding anti-dispersive terms of a special type. Previously, a method with similar properties was proposed only for the simplest one-dimensional linear advection equation. The modification of the scheme allows us to improve the dispersion properties of the advection for all Riemann invariants of the system of equations under consideration at once. The scheme is non-dissipative when the monotonization procedures are not used and is stable at Courant numbers CFL ≤ 1. The accuracy of the method and its order of convergence are shown in a series of solving the problem of advection of a wave modulated by a Gaussian on a sequence of condensing grids. The proposed method is planned to be used as a basis for constructing a CABARET scheme with improved dispersion properties for systems of nonlinear differential equations.

An Improved WENO-Z Scheme with Symmetry-Preserving Mapping

Since the classical weighted essentially non-oscillatory (WENO) scheme is proposed, various improved versions have been developed, and typical one is the WENO-Z scheme. Although better resolution is achieved, it is shown in this article that, the result of WENO-Z scheme suffers evident distortion in the long-time simulation of the linear advection equation. In order to fix the problem of WENO-Z, a symmetry-preserving mapping method is proposed in this article. In the original mapping method, the weight of each sub-stencil is used to map, which is demonstrated to cause asymmetric improvement about a discontinuity. This asymmetric improvement will lead to a distorted solution, more severe with longer output time. In the symmetry-preserving mapping method, a new variable related to the smoothness indicator is selected to map, which has the same ideal value for each sub-stencil. Using the new mapping method can not only fix the distortion problem of WENO-Z, but also improve the numerical resolution. Several benchmark problems are conducted to show the improved performance of the resultant scheme.

High order modified differential equation of the Beam–Warming method, II. The dissipative features

This paper is a continuation of [38]. The analysis of the modified partial differential equation (MDE) of the constant-wind-speed linear advection equation explicit difference scheme up to the eighth-order derivatives is presented. In this paper the authors focus on the dissipative features of the Beam–Warming scheme. The modified partial differential equation is presented in the so-called Π-form of the first differential approximation. The most important part of this form includes the coefficients μ (p) at the space derivatives. Analysis of these coefficients provides indications of the nature of the dissipative errors. A fragment of the stencil for determining the modified differential equation for the Beam–Warming scheme is included. The derived and presented coefficients μ (p) as well as the analysis of the dissipative features of this scheme on the basis of these coefficients have not been published so far.

An Improved WENO-Z Scheme with Symmetry-Preserving Mappin

Since the classical weighted essentially non-oscillatory (WENO) scheme is proposed, various improved versions have been developed, and a typical one is the WENO-Z scheme. Although better resolution is achieved, it is shown in this article that, the result of WENO-Z scheme suffers evident distortion in the long-time simulation of the linear advection equation. In order to fix the problem of WENO-Z scheme, a symmetry-preserving mapping method is proposed in this article. In the original mapping method, the weight of each substencil is used to map, which is demonstrated to cause asymmetric improvement about a discontinuity. This asymmetric improvement will lead to a distorted solution, more severe with longer output time. In the symmetry-preserving mapping method, a new variable related to the smoothness indicator is selected to map, which has the same ideal value for each substencil. Using the new mapping method can not only fix the distortion problem of WENO-Z scheme, but also improve the resolution. Several benchmark problems are conducted to show the improved performance of the resultant scheme.

p-norm regularization in variational data assimilation

<p> We introduce a new formulation of the 4DVAR objective function by using as a penalty term a p-norm with 1 < p < 2. So far, only the 2-norm, the 1-norm or a mixed of both have been considered as regularization term. This approach is motivated by the nature of the problems encountered in data assimilation, for which such a norm may be more suited to tackle the distribution of the variables. It also aims at making a compromise between the 2-norm that tends to oversmooth the solution or produce Gibbs oscillations, and the 1-norm that tends to "oversparcify" it, in addition to making the problem non-smooth.</p><p>The performance of the proposed technique are assessed for different p-values by twin experiments on a linear advection equation. The experiments are then conducted using two different true states in order to assess the performances of the p-norm regularized 4DVAR algorithm in sparse (rectangular function) and "almost" sparse cases (rectangular function with a smoother slope). In this setup, the background and the measurements noise covariance are known.</p><p>In order to minimize the 4DVAR objective function with a p-norm as a regularization term we use a gradient descent algorithm that requires the use of duality operators to work on a non-euclidean space. Indeed, Rn together with the p-norm (1 < p < 2) is a Banach space. Finally, to tune the regularization parameter appearing in the formulation of the objective function, we use the Morozov's discrepancy principle.</p>