Authors:Kaushik Mukherjee Pages: 167 - 189 Abstract: In this paper, we consider a class of singularly perturbed convection-diffusion boundary-value problems with discontinuous convection coefficient which often occur as mathematical models for analyzing shock wave phenomena in gas dynamics. In general, interior layers appear in the solutions of this class of problems and this gives rise to difficulty while solving such problems using the classical numerical methods (standard central difference or standard upwind scheme) on uniform meshes when the perturbation parameter ε is small. To achieve better numerical approximation in solving this class of problems, we propose a new hybrid scheme utilizing a layer-resolving piecewise-uniform Shishkin mesh and the method is shown to be ε-uniformly stable. In addition to this, it is proved that the proposed numerical scheme is almost second-order uniformly convergent in the discrete supremum norm with respect to the parameter ε. Finally, extensive numerical experiments are conducted to support the theoretical results. Further, the numerical results obtained by the newly proposed scheme are also compared with the hybrid scheme developed in the paper [Z.Cen, Appl. Math. Comput., 169(1): 689-699, 2005]. It shows that the current hybrid scheme exhibits a significant improvement over the hybrid scheme developed by Cen, in terms of the parameter-uniform order of convergence. PubDate: 2018-04-18 DOI: 10.3846/mma.2018.011 Issue No:Vol. 23, No. 2 (2018)
Authors:Krešimir Burazin, Jelena Jankov, Marko Vrdoljak Pages: 190 - 204 Abstract: We are interested in general homogenization theory for fourth-order elliptic equation describing the Kirchhoff model for pure bending of a thin solid symmetric plate under a transverse load. Such theory is well-developed for second-order elliptic problems, while some results for general elliptic equations were established by Zhikov, Kozlov, Oleinik and Ngoan (1979). We push forward an approach of Antoni´c and Balenovi´c (1999, 2000) by proving a number of properties of H-convergence for stationary plate equation. PubDate: 2018-04-18 DOI: 10.3846/mma.2018.012 Issue No:Vol. 23, No. 2 (2018)
Authors:Hongli Wang, Jianwei Yang Pages: 205 - 216 Abstract: The combined quasi-neutral and inviscid limit of the Navier-Stokes-Poisson-Korteweg system with density-dependent viscosity and cold pressure in the torus T3 is studied. It is shown that, for the well-prepared initial data, the global weak solution of the Navier-Stokes-Poisson-Korteweg system converges strongly to the strong solution of the incompressible Euler equations when the Debye length and the viscosity coefficient go to zero simultaneously. Furthermore, the rate of convergence is also obtained. PubDate: 2018-04-18 DOI: 10.3846/mma.2018.013 Issue No:Vol. 23, No. 2 (2018)
Authors:Said R. Grace, John R. Graef Pages: 217 - 226 Abstract: The authors establish some new criteria for the oscillation of solutions of second order nonlinear differential equations with a sublinear neutral term by reducing the equation to a linear one. Their results are illustrated with an example. PubDate: 2018-04-18 DOI: 10.3846/mma.2018.014 Issue No:Vol. 23, No. 2 (2018)
Authors:Mahmoud A. Zaky, Eid H. Doha, Taha M. Taha, Dumitru Baleanu Pages: 227 - 239 Abstract: To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation.In this paper, we develop Laguerre spectral collocation methods for solving variable-order fractional initial value problems on the half line. Specifically, we derive three-term recurrence relations to efficiently calculate the variable-order fractional integrals and derivatives of the modified generalized Laguerre polynomials, which lead to the corresponding fractional differentiation matrices that will be used to construct the collocation methods. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation methods. PubDate: 2018-04-18 DOI: 10.3846/mma.2018.015 Issue No:Vol. 23, No. 2 (2018)
Authors:Yujian Jiao, Tianjun Wang, Xiandong Shi, Wenjie Liu Pages: 240 - 261 Abstract: In this paper, we propose a mixed Jacobi-Fourier spectral method for solving the Fisher equation in a disc. Some mixed Jacobi-Fourier approximation results are established, which play important roles in numerical simulation of various problems defined in a disc. We use the generalized Jacobi approximation to simulate the singularity of solution at the regional center. This also simplifies the theoretical analysis and provides a sparse system. The stability and convergence of the proposed scheme are proved. Numerical results demonstrate the efficiency of this new algorithm and coincide well with the theoretical analysis. PubDate: 2018-04-18 DOI: 10.3846/mma.2018.016 Issue No:Vol. 23, No. 2 (2018)
Authors:Bin Han, Yukang Chen Pages: 262 - 286 Abstract: In [5], Chemin, Gallagher and Paicu proved the global regularity of solutions to the classical Navier-Stokes equations with a class of large initial data on T2 × R. This data varies slowly in vertical variable and has a norm which blows up as the small parameter ( represented by ǫ in the paper) tends to zero. However, to the best of our knowledge, the result is still unclear for the whole spaces R3. In this paper, we consider the generalized Navier-Stokes equations on Rn(n ≥ 3): ∂tu + u · ∇u + Dsu + ∇P = 0, div u = 0. For some suitable number s, we prove that the Cauchy problem with initial data of the form u0ǫ(x) = (v0h(xǫ), ǫ−1v0n(xǫ))T , xǫ = (xh, ǫxn)T , is globally well-posed for all small ǫ > 0, provided that the initial velocity profile v0 is analytic in xn and certain norm of v0 is sufficiently small but independent of ǫ. In particular, our result is true for the n-dimensional classical Navier-Stokes equations with n ≥ 4 and the fractional Navier-Stokes equations with 1 ≤ s < 2 in 3D. PubDate: 2018-04-18 DOI: 10.3846/mma.2018.017 Issue No:Vol. 23, No. 2 (2018)
Authors:Owe Axelsson, Maya Neytcheva, Zhao-Zheng Liang Pages: 287 - 308 Abstract: The recent development of the high performance computer platforms shows a clear trend towards heterogeneity and hierarchy. In order to utilize the computational power, particular attention must be paid to finding new algorithms or adjust existing ones so that they better match the HPC computer architecture. In this work we consider an alternative to classical time-stepping methods based on use of time-harmonic properties and discuss solution approaches that allow efficient utilization of modern HPC resources. The method in focus is based on a truncated Fourier expansion of the solution of an evolutionary problem. The analysis is done for linear equations and it is remarked on the possibility to use two- or multilevel mesh methods for nonlinear problems, which can enable further, even higher degree of parallelization. The arising block matrix system to be solved admits a two-by-two block form with square blocks, for which a very efficient preconditioner exists. It leads to tight eigenvalue bounds for the preconditioned matrix and, hence, to a very fast convergence of a preconditioned Krylov subspace or iterative refinement method. The analytical background is shown as well as some illustrating numerical examples. PubDate: 2018-04-18 DOI: 10.3846/mma.2018.018 Issue No:Vol. 23, No. 2 (2018)
Authors:Audrius Nečiūnas, Martynas Patašius, Rimantas Barauskas Pages: 309 - 326 Abstract: Conventional finite element method (FEM) is capable of obtaining wave solutions, but large discretized structures at high frequency require high computational resources, the computational domain can be reduced by combining FEM with analytical assumption for guided wave. Semi Analytical Finite Element (SAFE) formulation for immersed waveguide in perfect fluid is used for acquiring propagating wave modes as dynamic equilibrium states. Modes are solutions to eigenvalue problem and provide with important characteristic features of the guided waves – phase velocity, attenuation, wave structure, etc. The effect of surrounding leaky medium is modeled via traction boundary condition, which is based on assumption of the continuity of stresses at solid-fluid interface. The boundary condition causes wave attenuation due to energy leakage into outer medium. The derivation of the eigen-problem takes into account complex wavenumbers of leaky wave in fluid and guided wave in a three-dimensional waveguide. Linearization procedure for solving nonlinear eigenvalue problem is used. Dispersion relations for immersed waveguide with Rayleigh damping are obtained. The limits of applications of Rayleigh damping and convergence analysis of immersed waveguide model are discussed. PubDate: 2018-04-18 DOI: 10.3846/mma.2018.019 Issue No:Vol. 23, No. 2 (2018)
Authors:Harijs Kalis, Maksims Marinaki, Uldis Strautins, Maija Zake Pages: 327 - 343 Abstract: This paper deals with a simplified model taking into account the interplay of compressible, laminar, axisymmetric flow and the electrodynamical effects due to Lorentz force’s action on the combustion process in a cylindrical pipe. The combustion process with Arrhenius kinetics is modelled by a single step exothermic chemical reaction of fuel and oxidant. We analyze non-stationary PDEs with 6 unknown functions: the 3 components of velocity, density, concentration of fuel and temperature. For pressure the ideal gas law is used. For the inviscid flow approximation ADI method is used. Some numerical results are presented. PubDate: 2018-04-18 DOI: 10.3846/mma.2018.020 Issue No:Vol. 23, No. 2 (2018)
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