Central ADER schemes for hyperbolic conservation laws

In this paper we first briefly review the very high order ADER methods for solving hyperbolic conservation laws. ADER methods use high order polynomial reconstruction of the solution and upwind fluxes as the building block. They use a first order upwind Godunov and the upwind second order weighted average (WAF) fluxes. As well known the upwind methods are more accurate than central schemes. However, the superior accuracy of the ADER upwind schemes comes at a cost, one must solve exactly or approximately the Riemann problems (RP). Conventional Riemann solvers are usually complex and are not available for many hyperbolic problems of practical interest. In this paper we propose to use two central fluxes, instead of upwind fluxes, as the building block in ADER scheme. These are the monotone first order Lax-Friedrich (LXF) and the third order TVD flux. The resulting schemes are called central ADER schemes. Accuracy of the new schemes is established. Numerical implementations of the new schemes are carried out on the scalar conservation laws with a linear flux, nonlinear convex flux and non-convex flux. The results demonstrate that the proposed scheme, with LXF flux, is comparable to those using first and second order upwind fluxes while the scheme, with third order TVD flux, is superior to those using upwind fluxes. When compared with the state of art ADER schemes, our central ADER schemes are faster, more accurate, Riemann solver free, very simple to implement and need less computer memory. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented.     

Explicit finite difference schemes with extended stability for advection equations

In this study an explicit central difference approximation of the generalized leap-frog type is applied to the one- and two-dimensional advection equations. The stability of the considered numerical schemes is investigated and the scheme with the largest stable time step is found. For the linear and nonlinear advection equations numerical experiments with different schemes from the considered class are performed in order to evaluate the practical stability of the designed schemes.     

Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton-Jacobi equations

In this paper, a class of high-order central Hermite WENO (HWENO) schemes based on finite volume framework and staggered meshes is proposed for directly solving one- and two-dimensional Hamilton-Jacobi (HJ) equations. The methods involve the Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time. This work can be regarded as an extension of central HWENO schemes for hyperbolic conservation laws (Tao et al. J. Comput. Phys. 318, 222–251, 2016) which combine the central scheme and the HWENO spatial reconstructions and therefore carry many features of both schemes. Generally, it is not straightforward to design a finite volume scheme to directly solve HJ equations and a key ingredient for directly solving such equations is the reconstruction of numerical Hamiltonians to guarantee the stability of methods. Benefited from the central strategy, our methods require no numerical Hamiltonians. Meanwhile, the zeroth-order and the first-order moments of the solution are involved in the spatial HWENO reconstructions which is more compact compared with WENO schemes. The reconstructions are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. A collection of one- and two- dimensional numerical examples is performed to validate high resolution and robustness of the methods in approximating the solutions of HJ equations, which involve linear, nonlinear, smooth, non-smooth, convex or non-convex Hamiltonians.     

Split-step orthogonal spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions

Split-step orthogonal spline collocation (OSC) methods are proposed for one-, two-, and three-dimensional nonlinear Schrödinger (NLS) equations with time-dependent potentials. Firstly, the NLS equation is split into two nonlinear equations, and one or more one-dimensional linear equations. Commonly, the nonlinear subproblems could be integrated directly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. In this case, we propose three approximations by using quadrature formulae, but the split order is not reduced. Discrete-time OSC schemes are applied for the linear subproblems. In numerical experiments, many tests are carried out to prove the reliability and efficiency of the split-step OSC (SSOSC) methods. Solitons in one, two, and three dimensions are well simulated, and conservative properties and convergence rates are demonstrated. We also apply the ways of solving the nonlinear subproblems to the split-step finite difference (SSFD) methods and the time-splitting spectral (TSSP) methods, and the approximate ways still work well. Finally, we apply the SSOSC methods to solve some problems of Bose-Einstein condensates.     

Lax-Wendroff-type schemes of arbitrary order in several space dimensions

^{The second-order accurate Lax–Wendroff scheme is basedon the first three terms of a Taylor expansion in time in whichthe time derivatives are replaced by space derivatives usingthe governing evolution equations. The space derivatives arethen approximated by central differencing. In this paper, weextend this idea and truncate the Taylor expansion at an arbitraryorder. One main building block is the so-called Cauchy–Kovalevskayaprocedure to replace all the time derivatives by space derivativeswhich can be formulated for a general system of linear equationswith arbitrary order and in two- or three-space dimensions.The linear case is the main focus of this paper because theproposed high-order schemes are good candidates for the approximationof linear wave motion over long distances and times with importantapplications in aeroacoustics and electromagnetics. The stabilityand the efficiency of Lax–Wendroff-type schemes are examined.The numerical results are compared with a standard scheme foraeroacoustical applications with respect to their quality andthe computational effort. The extensions of the schemes to generalgrids, nonconstant and nonlinear cases are alsoaddressed.}     

Integrability and dark soliton solutions for a high-order variable coefficients nonlinear Schrödinger equation

Under investigation in this paper is the integrability and dark soliton solutions for a fifth-order variable coefficients nonlinear Schrödinger equation, which is used in an inhomogeneous optical fiber. Bilinear forms, Lax pair and infinitely-many conservation laws are obtained under an integrable constraint. Dark one-, two- and N-soliton solutions are constructed via the Hirota bilinear method.     

Revisit of Semi-Implicit Schemes for Phase-Field Equations

It is a very common practice to use semi-implicit schemes in various computations, which treat selected linear terms implicitly and the nonlinear terms explicitly. For phase-field equations, the principal elliptic operator is treated implicitly to reduce the associated stability constraints while the nonlinear terms are still treated explicitly to avoid the expensive process of solving nonlinear equations at each time step. However, very few recent numerical analysis is relevant to semi-implicit schemes, while "stabilized" schemes have become very popular. In this work, we will consider semi-implicit schemes for the Allen-Cahn equation with $general$ $potential$ function. It will be demonstrated that the maximum principle is valid and the energy stability also holds for the numerical solutions. This paper extends the result of Tang & Yang (J. Comput. Math., 34(5) (2016), pp. 471-481), which studies the semi-implicit scheme for the Allen-Cahn equation with $polynomial$ $potentials$.     

Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).     

Stability of Manifold-Valued Subdivision Schemes and Multiscale Transformations

Linear subdivision schemes can be adapted in various ways so as to operate in nonlinear geometries such as Lie groups or Riemannian manifolds. It is well known that along with a linear subdivision scheme a multiscale transformation is defined. Such transformations can also be defined in a nonlinear setting. We show the stability of such nonlinear multiscale transforms. To do this we introduce a new kind of proximity condition which bounds the difference of the differential of a nonlinear subdivision scheme and a linear one. It turns out that—unlike the generic nonlinear case and modulo some minor technical assumptions—in the manifold-valued setting, convergence implies stability of the nonlinear subdivision scheme and associated nonlinear multiscale transformations.