In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional int...In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.展开更多
In this paper,we present a Runge-Kutta Discontinuous Galerkin(RKDG)method for solving the two-dimensional ideal compressible magnetohydrodynamics(MHD)equations under the Lagrangian framework.The fluid part of the idea...In this paper,we present a Runge-Kutta Discontinuous Galerkin(RKDG)method for solving the two-dimensional ideal compressible magnetohydrodynamics(MHD)equations under the Lagrangian framework.The fluid part of the ideal MHD equations along with z-component of the magnetic induction equation are discretized using a DG method based on linear Taylor expansions.By using the magnetic fluxfreezing principle which is the integral form of the magnetic induction equation of the ideal MHD,an exactly divergence-free numerical magnetic field can be obtained.The nodal velocities and the corresponding numerical fluxes are explicitly calculated by solving multidirectional approximate Riemann problems.Two kinds of limiter are proposed to inhibit the non-physical oscillation around the shock wave,and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of the rotor problems.This Lagrangian RKDG method conserves mass,momentum,and total energy.Several numerical tests are presented to demonstrate the accuracy and robustness of the proposed scheme.展开更多
文摘In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.
基金supported by National Natural Science Foundation of China(12071046,11671049,91330107,11571002 and 11702028)China Postdoctoral Science Foundation(2020TQ0013).
文摘In this paper,we present a Runge-Kutta Discontinuous Galerkin(RKDG)method for solving the two-dimensional ideal compressible magnetohydrodynamics(MHD)equations under the Lagrangian framework.The fluid part of the ideal MHD equations along with z-component of the magnetic induction equation are discretized using a DG method based on linear Taylor expansions.By using the magnetic fluxfreezing principle which is the integral form of the magnetic induction equation of the ideal MHD,an exactly divergence-free numerical magnetic field can be obtained.The nodal velocities and the corresponding numerical fluxes are explicitly calculated by solving multidirectional approximate Riemann problems.Two kinds of limiter are proposed to inhibit the non-physical oscillation around the shock wave,and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of the rotor problems.This Lagrangian RKDG method conserves mass,momentum,and total energy.Several numerical tests are presented to demonstrate the accuracy and robustness of the proposed scheme.
