In this article,a robust,effective,and scale-invariant weighted compact nonlinear scheme(WCNS)is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes.The new scheme achieves an...In this article,a robust,effective,and scale-invariant weighted compact nonlinear scheme(WCNS)is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes.The new scheme achieves an essentially non-oscillatory approximation of a discontinuous function(ENO-property),a scaleinvariant property with an arbitrary scale of a function(Si-property),and an optimal order of accuracy with smooth function regardless of the critical point(Cp-property).The classical WCNS-Z/D schemes do not satisfy Si-property intrinsically,which is caused by a loss of sub-stencils’adaptivity in the nonlinear interpolation of a discontinuous function when scaled by a small scale factor.A new nonlinear weight is devised by using an average of the function values and the descaling function,providing the new WCNS schemes(WCNS-Zm/Dm)with many attractive properties.The ENO-property,Si-property and Cp-property of the new WCNS schemes are validated numerically.Results show that the WCNS-Zm/Dm schemes satisfy the ENO-property and Si-property,while only the WCNS-Dm scheme satisfies the Cp-property.In addition,the Gaussian wave problem is solved by using successively refined grids to verify that the optimal order of accuracy of the new schemes can be achieved.Several one-dimensional shock tube problems,and two-dimensional double Mach reflection(DMR)problem and the Riemann IVP problem are simulated to illustrate the ENOproperty and Si-property of the scale-invariant WCNS-Zm/Dm schemes.展开更多
In this study,we investigate the seventh-order nonlinear Caputo time-fractional KdV equation.The suggested model's solutions,which have a series form,are obtained using the hybrid ZZ-transform under the aforementi...In this study,we investigate the seventh-order nonlinear Caputo time-fractional KdV equation.The suggested model's solutions,which have a series form,are obtained using the hybrid ZZ-transform under the aforementioned fractional operator.The proposed approach combines the homotopy perturbation method(HPM)and the ZZ-transform.We consider two specific examples with suitable initial conditions and find the series solution to test their applicability.To demonstrate the utility of the presented technique,we explore its applications to the fractional Sawada–Kotera–Ito problem and the Lax equation.We observe the impact of a few fractional orders on the wave solution evolution for the problems under consideration.We provide the efficiency and reliability of the ZZHPM by calculating the absolute error between the series solution and the exact solution of both the Sawada–Kotera–Ito and Lax equations.The convergence and uniqueness of the solution are portrayed via fixed-point theory.展开更多
基金Project supported by the National Key Research and Development Project of China(2016YFA0401200)the National Natural Science Foundation of China(11301525)~~
基金supported by the Hunan Provincial Natural Science Foundation of China(No.2022JJ40539)National Natural Science Foundation of China(No.11972370)National Key Project(No.GJXM92579).
文摘In this article,a robust,effective,and scale-invariant weighted compact nonlinear scheme(WCNS)is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes.The new scheme achieves an essentially non-oscillatory approximation of a discontinuous function(ENO-property),a scaleinvariant property with an arbitrary scale of a function(Si-property),and an optimal order of accuracy with smooth function regardless of the critical point(Cp-property).The classical WCNS-Z/D schemes do not satisfy Si-property intrinsically,which is caused by a loss of sub-stencils’adaptivity in the nonlinear interpolation of a discontinuous function when scaled by a small scale factor.A new nonlinear weight is devised by using an average of the function values and the descaling function,providing the new WCNS schemes(WCNS-Zm/Dm)with many attractive properties.The ENO-property,Si-property and Cp-property of the new WCNS schemes are validated numerically.Results show that the WCNS-Zm/Dm schemes satisfy the ENO-property and Si-property,while only the WCNS-Dm scheme satisfies the Cp-property.In addition,the Gaussian wave problem is solved by using successively refined grids to verify that the optimal order of accuracy of the new schemes can be achieved.Several one-dimensional shock tube problems,and two-dimensional double Mach reflection(DMR)problem and the Riemann IVP problem are simulated to illustrate the ENOproperty and Si-property of the scale-invariant WCNS-Zm/Dm schemes.
文摘In this study,we investigate the seventh-order nonlinear Caputo time-fractional KdV equation.The suggested model's solutions,which have a series form,are obtained using the hybrid ZZ-transform under the aforementioned fractional operator.The proposed approach combines the homotopy perturbation method(HPM)and the ZZ-transform.We consider two specific examples with suitable initial conditions and find the series solution to test their applicability.To demonstrate the utility of the presented technique,we explore its applications to the fractional Sawada–Kotera–Ito problem and the Lax equation.We observe the impact of a few fractional orders on the wave solution evolution for the problems under consideration.We provide the efficiency and reliability of the ZZHPM by calculating the absolute error between the series solution and the exact solution of both the Sawada–Kotera–Ito and Lax equations.The convergence and uniqueness of the solution are portrayed via fixed-point theory.
