Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem

We present a new conservative semi-Lagrangian finite difference weighted essentially non-oscillatory scheme with adaptive order. This is an extension of the conservative semi-Lagrangian (SL) finite difference WENO scheme in [Qiu and Shu, JCP, 230 (4) (2011), pp. 863-889], in which linear weights in SL WENO framework were shown not to exist for variable coefficient problems. Hence, the order of accuracy is not optimal from reconstruction stencils. In this paper, we incorporate a recent WENO adaptive order (AO) technique [Balsara et al., JCP, 326 (2016), pp. 780-804] to the SL WENO framework. The new scheme can achieve an optimal high order of accuracy, while maintaining the properties of mass conservation and non-oscillatory capture of solutions from the original SL WENO. The positivity-preserving limiter is further applied to ensure the positivity of solutions. Finally, the scheme is applied to high dimensional problems by a fourth-order dimensional splitting. We demonstrate the effectiveness of the new scheme by extensive numerical tests on linear advection equations, the Vlasov-Poisson system, the guiding center Vlasov model as well as the incompressible Euler equations.     

A Nonlinear Finite Volume Scheme Preserving Maximum Principle for Diffusion Equations

In this paper we propose a new nonlinear cell-centered finite volume scheme on general polygonal meshes for two dimensional anisotropic diffusion problems, which preserves discrete maximum principle (DMP). The scheme is based on the so-called diamond scheme with a nonlinear treatment on its tangential flux to obtain a local maximum principle (LMP) structure. It is well-known that existing DMP preserving diffusion schemes suffer from the fact that auxiliary unknowns should be presented as a convex combination of primary unknowns. In this paper, to get rid of this constraint a nonlinearization strategy is introduced and it requires only a second-order accurate approximation for auxiliary unknowns. Numerical results show that this scheme has second-order accuracy, preserves maximum and minimum for solutions and is conservative.     

High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations

In this paper, we develop two finite difference weighted essentiallynon-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving theDegasperis-Procesi (DP) and $mu$-Degasperis-Procesi ($mu$DP) equations, which containnonlinear high order derivatives, and possibly peakon solutions or shock waves. Byintroducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-ellipticsystem, and the $mu$DP equation as a first order system. Then we choose a linear finitedifference scheme with suitable order of accuracy for the auxiliary variable(s), andtwo finite difference WENO schemes with unequal-sized sub-stencils for the primalvariable. One WENO scheme uses one large stencil and several smaller stencils, andthe other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENOscheme which uses several small stencils of the same size to make up a big stencil, bothWENO schemes with unequal-sized sub-stencils are simple in the choice of the stenciland enjoy the freedom of arbitrary positive linear weights. Another advantage is thatthe final reconstructed polynomial on the target cell is a polynomial of the same degree as the polynomial over the big stencil, while the classical finite difference WENOreconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatoryproperties of the proposed schemes.     

A Sufficient and Necessary Condition of the Existence of WENO-Like Linear Combination for Finite Difference Schemes

In the finite difference WENO (weighted essentially non-oscillatory) method, the final scheme on the whole stencil was constructed by linear combinations ofhighest order accurate schemes on sub-stencils, all of which share the same total countof grid points. The linear combination method which the original WENO applied wasgeneralized to arbitrary positive-integer-order derivative on an arbitrary (uniform ornon-uniform) mesh, still applying finite difference method. The possibility of expressing the final scheme on the whole stencil as a linear combination of highest order accurate schemes on WENO-like sub-stencils was investigated. The main results include:(a) the highest order of accuracy a finite difference scheme can achieve and (b) a sufficient and necessary condition that the linear combination exists. This is a sufficientand necessary condition for all finite difference schemes in a set (rather than a specificfinite difference scheme) to have WENO-like linear combinations. After the proofsof the results, some remarks on the WENO schemes and TENO (targeted essentiallynon-oscillatory) schemes were given.     

A Higher Order Interpolation Scheme of Finite Volume Method for Compressible Flow on Curvilinear Grids

A higher order interpolation scheme based on a multi-stage BVD (Boundary Variation Diminishing) algorithm is developed for the FV (Finite Volume) method on non-uniform, curvilinear structured grids to simulate the compressible turbulent flows. The designed scheme utilizes two types of candidate interpolants including a higher order linear-weight polynomial as high as eleven and a THINC (Tangent of Hyperbola for INterface Capturing) function with the adaptive steepness. We investigate not only the accuracy but also the efficiency of the methodology through the cost efficiency analysis in comparison with well-designed mapped WENO (Weighted Essentially Non-Oscillatory) scheme. Numerical experimentation including benchmark broadband turbulence problem as well as real-life wall-bounded turbulent flows has been carried out to demonstrate the potential implementation of the present higher order interpolation scheme especially in the ILES (Implicit Large Eddy Simulation) of compressible turbulence.     

High Order Finite Difference Hermite WENO Fixed-Point Fast Sweeping Method for Static Hamilton-Jacobi Equations

In this paper, we combine the nonlinear HWENO reconstruction in [43] andthe fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the staticHamilton-Jacobi equations in a novel HWENO framework recently developed in [22].The proposed HWENO frameworks enjoys several advantages. First, compared withthe traditional HWENO framework, the proposed methods do not need to introduceadditional auxiliary equations to update the derivatives of the unknown function $phi$.They are now computed from the current value of $phi$ and the previous spatial derivatives of $phi$. This approach saves the computational storage and CPU time, which greatlyimproves the computational efficiency of the traditional HWENO scheme. In addition,compared with the traditional WENO method, reconstruction stencil of the HWENOmethods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping methodis used to update the numerical approximation. It is an explicit method and doesnot involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves someknown issues, including that the iterative number is very sensitive to the parameter $ε$ used in the nonlinear weights, as observed in previous studies. Finally, to furtherreduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate thegood performance of the proposed fixed-point fast sweeping HWENO methods.     

Arbitrarily High-Order (Weighted) Essentially Non-Oscillatory Finite Difference Schemes for Anelastic Flows on Staggered Meshes

We propose a WENO finite difference scheme to approximate anelastic flows, and scalars advected by them, on staggered grids. In contrast to existing WENO schemes on staggered grids, the proposed scheme is designed to be arbitrarily high-order accurate as it judiciously combines ENO interpolations of velocities with WENO reconstructions of spatial derivatives. A set of numerical experiments are presented to demonstrate the increase in accuracy and robustness with the proposed scheme, when compared to existing WENO schemes and state-of-the-art central finite difference schemes.     

A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model

In this paper we propose and analyze a (temporally) third order accuratebackward differentiation formula (BDF) numerical scheme for the no-slope-selection(NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectraldiscretization in space. The surface diffusion term is treated implicitly, while the nonlinear chemical potential is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupontregularization term, in the form of $−A∆t^2∆^2_N(u^{n+1}−u^n)$, is added in the numericalscheme. A careful energy stability estimate, combined with Fourier eigenvalue analysis, results in the energy stability in a modified version, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, theoptimal rate convergence analysis and error estimate are derived in details, in the $ℓ^∞(0,T;ℓ^2)∩ℓ^2(0,T;H^2_h)$ norm, with the help of a linearized estimate for the nonlinear error terms. Some numerical simulation results are presented to demonstrate theefficiency of the numerical scheme and the third order convergence. The long timesimulation results for $ε = 0.02$ (up to $T = 3×10^5$) have indicated a logarithm law forthe energy decay, as well as the power laws for growth of the surface roughness andthe mound width. In particular, the power index for the surface roughness and themound width growth, created by the third order numerical scheme, is more accuratethan those produced by certain second order energy stable schemes in the existingliterature.     

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [11] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In addition, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.