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Weighted essentially non-oscillatory (WENO) methods can simultaneously provide the high order of accuracy, high bandwidth-resolving efficiency, and shock-capturing capability required for the detailed simulation of compressible turbulence. However, rigorous analysis of the actual versus theoretical error properties of these non-linear numerical methods is difficult. We use a bandwidth-optimized WENO scheme to conduct direct numerical simulations of two- and three-dimensional decaying isotropic turbulence, and we evaluate the performance of quantitative indicators of local WENO adaptation behavior within the resulting flow fields. One aspect of this assessment is the demarcation of shock-containing and smooth regions where the WENO method should, respectively, engage its adaptation mechanism and revert to its linear optimal stencil. Our results show that these indicators, when synthesized properly, can provide valuable quantitative information suitable for statistical characterization.

In this paper, evolution and visualization of the existence of saddle points of nonlinear functionals or multi-variable functions in finite dimensional spaces are presented. New algorithms are developed based on the mountain pass lemma and link thery in nonlinear analysis. Further more, a simple comparison of the steepest descent algorithm and the genetic algorithm is given. The process of the saddle point finding is visualised in an inteactive graphical interface.

A pseudo-spectral (or collocation) approximation of the unsteady Stokes equations is presented. Using the Uzawa algorithm the spectral system is decoupled into Helmholtz equations for the velocity components and an equation with the Pseudo-Laplacian for the pressure. In order to avoid spurious modes the pressure is approximated with lower order (two degrees lower) polynomials than the velocity. Only one grid (no staggered grids) with the standard Chebyshev Gauss-Lobatto nodes is used. Here we further compare our treatment with a Neumann boundary value problem for the pressure. The highly improved accuracy of our method becomes obvious. In the time discretization a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure. It is numerically shown that a stable third order method in time can be achieved.

We present a numerical method for the simulation of binary alloys. We make use of the level-set method to capture the evolution of the solidification front and of an adaptive mesh refinement framework based on non-graded quadtree grids to efficiently capture the multiscale nature of the alloys’ concentration profile. In addition, our approach is based on a sharp treatment of the boundary conditions at the solidification front. We apply this algorithm to the solidification of an Ni–Cu alloy and report results that agree quantitatively with theoretical analyses. We also apply this algorithm to show that solidification mechanism maps predicting growth regimes as a function of tip velocities and thermal gradients can be accurately computed with this method; these include the important transitions from planar to cellular to dendritic regimes.

We propose an accurate algorithm for a novel sum-of-exponentials (SOE) approximation of kernel functions, and develop a fast algorithm for convolution quadrature based on the SOE, which allows an order N calculation for N time steps of approximating a continuous temporal convolution integral. The SOE method is constructed by a combination of the de la Vallée-Poussin sum for a semi-analytical exponential expansion of a general kernel, and a model reduction technique to minimize the number of exponentials under a given error tolerance. We employ the SOE expansion for the finite part of the splitting convolution kernel so that the convolution integral can be solved as a system of ordinary differential equations. We show that the SOE method works for general kernels with controllable upperbound of positive exponents. Numerical analysis is provided for both the SOE method and the SOE-based convolution quadrature. Numerical results on different kernels demonstrate attractive performance on both accuracy and efficiency of the proposed method.

We propose a homotopy method to solve the quadratic two-parameter eigenvalue problems, which arise frequently in the analysis of the asymptotic stability of the delay differential equation. Our method does not require to form coupled generalized eigenvalue problems with Kronecker product type coefficient matrices and thus can avoid the increasing of the computational cost and memory storage. Numerical results and the applications in the delay differential equations are presented to illustrate the effectiveness and efficiency of our method. It appears that our method tends to be more effective than the existing methods in terms of speed, accuracy and memory storage as the problem size grows.

We propose and analyze a BDF2 scheme with variable time steps for the Cahn–Hilliard equation with a logarithmic Flory–Huggins energy potential. The lumped mass method is adopted in the space discretization to ensure that the proposed scheme is uniquely solvable and positivity-preserving. Especially, a new second order viscous regularization term is added at the discrete level to guarantee the energy dissipation property. Furthermore, the energy stability is derived by a careful estimate under the condition that $$r\le r_{\max }$$ . To estimate the spatial and temporal errors separately, a spatially semi-discrete scheme is proposed and a new elliptic projection is introduced, and the super-closeness between this projection and the Ritz projection of the exact solution is attained. Based on the strict separation property of the numerical solution obtained by using the technique of combining the rough and refined error estimates, the convergence analysis in $$l^{\infty }(0,T;L_h^2(\varOmega ))$$ norm is established when $$\tau \le Ch$$ by using the technique of the DOC kernels. Finally, several numerical experiments are carried out to validate the theoretical results.

We investigate a hybrid Fourier-Continuation (FC) method (Bruno and Lyon, J Comput Phys 229:2009–2033, 2010) and fifth order characteristic-wise weighted essentially non-oscillatory (WENO) finite difference scheme for solving system of hyperbolic conservation laws on a uniformly discretized Cartesian domain. The smoothness of the solution is measured by the high order multi-resolution algorithm by Harten (J Comput Phys 49:357–393, 1983) at each grid point in a single-domain framework (Costa and Don, J Comput Appl Math 204(2):209–218, 2007) (Hybrid), as opposed to each subdomain in a multi-domain framework (Costa et al., J Comput Phys 224(2):970–991, 2007; Shahbazi et al., J Comput Phys 230:8779–8796, 2011). The Hybrid scheme conjugates a high order shock-capturing WENO-Z5 (nonlinear) scheme (Borges et al., J Comput Phys 227:3101–3211, 2008) in non-smooth WENO stencils with an essentially non-dissipative and non-dispersive FC (linear) method in smooth FC stencils, yielding a high fidelity scheme for applications containing both discontinuous and complex smooth structures. Several critical and unique numerical issues in an accurate and efficient implementation (such as reasonable choice of parameters, singular value decomposition, fast Fourier transform, symmetry preservation, and overlap zone) of the FC method, due to a dynamic spatial and temporal change in the size of data length in smooth FC stencils in a single-domain framework, will be illustrated and addressed. The accuracy and efficiency of the Hybrid scheme in solving one and two dimensional system of hyperbolic conservation laws is demonstrated with several classical examples of shocked flow, such as the one dimensional Riemann initial value problems (123, Sod and Lax), the Mach 3 shock–entropy wave interaction problem with a small entropy sinusoidal perturbation, the Mach 3 shock–density wave interaction problem, and the two dimensional Mach 10 double Mach reflection problem. For a sufficiently large problem size, a factor of almost two has been observed in the speedup of the Hybrid scheme over the WENO-Z5 scheme.

We analyze space semi-discretizations of linear, second-order wave equations by discontinuous Galerkin methods in polygonal domains where solutions exhibit singular behavior near corners. To resolve these singularities, we consider two families of locally refined meshes: graded meshes and bisection refinement meshes. We prove that for appropriately chosen refinement parameters, optimal asymptotic rates of convergence with respect to the total number of degrees of freedom are obtained, both in the energy norm errors and the $$\mathcal {L}^2$$ -norm errors. The theoretical convergence orders are confirmed in a series of numerical experiments which also indicate that analogous results hold for incompatible data which is not covered by the currently available regularity theory.