High Order Optimized WENO Schemes For Some Partial Differential Equations

WENO schemes are high order numerical methods for solving PDEs which may contain discontinuities,sharp gradient regions and other complex solution structures.The main idea of WENO methodology is to form a weighted combination of several local reconstructions based on different stencils and use it as the final WENO reconstruction.Therefore,WENO schemes have the ability to achieve high order accuracy in smooth regions while maintaining sharp and essentially monotone shock transitions.This dissertation is devoted to construct novel WENO schemes for some important nonlinear partial differential equations.Meanwhile,we investigate the responding convergence,maximum-principle-preserving(MPP)and positivitypreserving(PP)properties of proposed algorithms theoretically and numerically.Main contributions of this dissertation are as follows:1.A high order coupling algorithm is proposed for the DP equation.The DP equation is split into a hyperbolic equation and an elliptic equation.For the hyperbolic equation,we use an optimized finite difference WENO scheme.New smoothness measurement is presented to approximate the typical shockpeakon structure in the solution of the DP equation,which evidently reduces the dissipation arising from the discontinuity.For the elliptic equation,the Fourier pseudospectral method(FPM)is employed to discretize the high order derivatives.Due to the combination of the WENO reconstruction and the FPM,the splitting method shows an excellent ability in capturing the formation and propagation of shockpeakon solutions.The numerical simulations for different solutions of the DP equation are conducted to illustrate the high accuracy and capability of the method.2.A novel third order WENO scheme is proposed for hyperbolic equations.We modify the construction of the third order finite volume WENO scheme on triangular meshes and present a simplified WENO(SWENO)scheme.The novelty of the SWENO scheme is the less complexity and lower computational cost when deciding the smoothest stencil through a simple mechanism.The LU decomposition with iterative refinement is adopted to implement illconditioned interpolation matrices and improves the stability of the SWENO scheme efficiently.Besides,a scaling technique is employed to circument the growth of condition numbers as mesh refined.However,weak oscillations still appear when the SWENO scheme deals with complex low density equations.In order to guarantee the MPP/PP property,we apply a scaling limiter to modify the reconstruction polynomial without the loss of accuracy.A special procedure is designed to prove this property theoretically.Finally,numerical examples for one-and two-dimensional problems are presented to verify the good performance,maximum principle preserving,essentially non-oscillation and high resolution of the proposed scheme.