In this paper we introduce a conforming discontinuous Galerkin(CDG)finite element method for solving the heat equation.Unlike the rest of discontinuous Galerkin(DG)methods,the numerical-flux {▽uh·n} is not introduced to the computation in the CDG method.Addi-tionally,the numerical-trace {uh} is not the average(uh|T1+uh|T2)/2(or some other simple average used in other DG methods),but a lifted Pk+1 polynomial from the Pk solution uh on nearby four triangles in 2D,or eight tetrahedra in 3D.We show a two-order supercon-vergence in space approximation when using the CDG method with a backward Euler time discretization,on triangular and tetrahedral meshes,for solving the heat equation.Numerical tests are reported which confirm the theory.
In this paper, we propose a numerical approach for the fractional reaction sub-diffusion equation with a Caputo-Hadamard derivative of fractional order α∈ (0,1) , whose solutions typically exhibit singular behavior at the initial time. The approach involves the use of an L2-type discrete operator of order 3-α to approximate the fractional derivative on nonuniform temporal meshes, while a standard second-order difference method is employed for uniform spatial meshes. Detailed discussions are presented on the truncation errors and coefficients of the proposed discrete fractional derivative operator. The stability as well as the accuracy of the resulting numerical scheme is rigorously analyzed on a unique nonuniform mesh. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.
We aim to propose a Riemann-problem-solver-free nonstaggered central difference scheme (NCDS), which is well-balanced and positivity-preserving for two-dimensional shallow water flows over nonflat bottom topography with wet-dry fronts. The well-balanced property of the NCDS for the two-dimensional shallow water equations at wet-dry fronts is still unclear. The NCDS is “generously multidimensional” and consists of three steps: a forward step, a corrector step, and a backward step. Each step needs to guarantee the water depth to be nonnegative and retain the lake at rest. The key ingredient of the NCDS is the backward step. It is very nontrivial and technical to obtain both the well-balanced and positivity-preserving properties when the computational domain contains wet-dry fronts for the NCDS. The traditional NCDS fails to retain the lake at rest when the computational domain contains wet-dry fronts. We introduce a novel backward step, which is well-balanced and positivity-preserving, to recover numerical solutions on nonstaggered cells. Finally, we show several numerical results of two-dimensional shallow water equations with wet-dry fronts to verify these properties.
We investigate the inverse scattering transform for the focusing nonlinear Schrödinger (NLS) equation with a particular class of nonvanishing boundary conditions (NVBCs), especially in the case of reflectionless potentials that give rise to a transmission coefficient with an arbitrary finite number N pairs of higher-order poles. The inverse problem is characterized in terms of a 2× 2 matrix Riemann-Hilbert (RH) problem equipped with several residue conditions at N pairs of higher-order poles. In the reflectionless case, we point out that the N-multipole soliton solutions including higher-order Kuznetsov-Ma breathers and Akhmediev breathers can be reconstructed by a linear algebraic system. Furthermore, we verify these special solutions by numerical simulations and display their density structures, also derive some Peregrine solitons by choosing appropriate parameters and taking limits.
Reaction-diffusion equations model various biological, physical, sociological, and environmental phenomena. Often, numerical simulations are used to understand and discover the dynamics of such systems. Following the extension of the nonlinear Lyapunov theory applied to some class of reaction-diffusion partial differential equations (PDEs), we develop the first fully discrete Lyapunov discretizations that are consistent with the stability properties of the continuous parabolic reaction-diffusion models. The proposed framework provides a systematic procedure to develop fully discrete schemes of arbitrary order in space and time for solving a broad class of equations equipped with a Lyapunov functional. The new schemes are applied to solve systems of PDEs, which arise in epidemiology and oncolytic M1 virotherapy. The new computational framework provides physically consistent and accurate results without exhibiting scheme-dependent instabilities and converging to unphysical solutions. The proposed approach represents a capstone for developing efficient, robust, and predictive technologies for simulating complex phenomena.
