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We are promoting the use of variational principles as some generalizing idea that allows us to create consistent approximations of mathematical models. In this paper, we formulate a computing technology to solve direct and inverse problems of atmospheric dynamics and chemistry. For the implementation, we use functional decomposition methods, splitting techniques, and integrating factors designed as the solutions of some adjoint problems. One of the main purposes of the paper is to relate the theory of approximation with the concept of Eulerian integrating factors that plays the fundamental role in the theory of differential equations. We demonstrate the utility and versatility of the concept applying it to solve the systems of differential equations of convection–diffusion (with dominant convection) and stiff systems of kinetic equations. To identify the prospects of our approach, we briefly introduce how to use integrating factors in the case of unstructured grids. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

In this paper, efficient numerical schemes are proposed for solving the fractional water wave models that describe the propagation of surface water wave. By using the weighted and shifted Grünwald–Letnikov (WSGL) formula to approximate the nonlocal fractional operators, we design a series of second order accurate difference schemes for the considered models. The existence, stability and convergence of numerical solutions of the proposed numerical schemes are established rigorously. The analysis shows that the proposed numerical schemes are unconditionally stable with second order accuracy for both temporal and spatial discretizations. Several numerical results are provided to verify the efficiency and accuracy of our theoretical analysis. [ABSTRACT FROM AUTHOR]

The goal of this paper is to obtain a numerical approximation of the effective Hamiltonian for a system of PDEs deriving from a second order discrete model for traffic flow. We will propose an explicit and an implicit discretization for this effective Hamiltonian and we will provide the corresponding error estimates of Crandall–Lions type. Finally, we will also present some numerical simulations. [ABSTRACT FROM AUTHOR]

This paper proposes a new approximation method for conic sections by quartic Bézier curves with G 1 end-point continuity. We give an upper bound on the Hausdorff distance between the conic section and the quartic Bézier approximation curve, and also show that the approximation order is eight. Moreover, using the subdivision scheme at the shoulder point of the conic section, the resulting approximation curve has G 2 (curvature)-continuity with the conic section at the endpoints. Finally, we prove that our approximation has a smaller error bound than previous quartic Bézier approximations, and also present some numerical examples to demonstrate the validity and effectiveness of our method. [ABSTRACT FROM AUTHOR]

In the present paper we consider a 1D Poisson model characterized by the presence of an interface, where a transmission condition arises due to jumps of the coefficients. We aim at studying finite element methods with meshes not fitting such an interface. It is well known that when the mesh does not fit the material discontinuities the resulting scheme provides in general lower order accurate solutions. We focus on so-called embedded approaches, frequently adopted to treat fluid–structure interaction problems, with the aim of recovering higher order of approximation also in presence of non fitting meshes; we implement several methods inspired by: the Immersed Boundary method, the Fictitious Domain method, and the Extended Finite Element method. In particular, we present four formulations in a comprehensive and unified format, proposing several numerical tests and discussing their performance. Moreover, we point out issues that may be encountered in the generalization to higher dimensions and we comment on possible solutions. [ABSTRACT FROM AUTHOR]