| In this thesis,we propose a type of efficient weighted essentially non-oscillatory(WENO)schemes for solving hyperbolic conservation laws,including Hermite WENO(HWENO)and hybrid WENO schemes,and applied them to simulate multi-medium flows.The main research results are as follows:Firstly,we present two hybrid finite volume HWENO schemes.Unlike other HWENO schemes,we bring the thought of limiter for discontinuous Galerkin method to modify the first-order moment near discontinuities by HWENO method for the first time,and directly use linear approximation in smooth regions.The first advantage is that the modification for the first-order moment can overcome the oscillations better,making the schemes be more robust,and the second one is that the schemes are efficient.We choose two different methods for HWENO reconstruction,in which the first method needs to calculate the linear weights based on the meshes and reconstructed points,while the second can use the artificial positive linear weights(their sum being one).Compared with the first one,the second has higher order accuracy in two-dimensional case under the condition of using less candidate stencils.Secondly,we present a hybrid finite difference WENO scheme.The scheme chooses WENO reconstruction or linear approximation automatically based on the locations of the extreme points of the reconstruction polynomial,and employs a new WENO reconstruction method,where the artificial positive linear weights(their sum being one)can be used.The first advantage is that the scheme is more efficient compared with the original WENO scheme,and the second one is that the scheme is robust,i.e.,it can also be utilized to simulate the rather extreme tests without any additional positivity-preserving technique.Thirdly,we present a modified finite difference HWENO scheme,in which we first modify the derivative of the solution by HWENO interpolation,then,employ HWENO method to reconstruct the numerical flux.Compared with the original finite difference HWENO scheme,the first advantage is that the scheme can apply lager CFL number and does not need to use any additional positivity-preserving methodology,and the second one is the scheme has higher order accuracy in two-dimensional case under the condition of using the same approximation stencils and information.Fourthly,we apply the HWENO method to simulate multi-medium flows for the first time,in which we use the modified ghost fluid method to transform a multi-medium flow problem to multi single-medium problems,which would be solved later by the robust and efficient finite difference HWENO scheme.Compared with the classical WENO method,the HWENO method needs less ghost point and boundary point.Finally,the numerical tests are presented to illustrate the robustness and high efficiency of the WENO and HWENO schemes.Compared with the WENO scheme with the same order accuracy,the HWENO schemes have more compact reconstruction stencils,less numerical errors and higher resolution. |
