Zhanjing Tao

Zhanjing Tao
Jilin University | JUT · Department of Mathematics

PhD

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20
Publications
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136
Citations

Publications

Publications (20)
Article
Full-text available
In this paper, a class of high-order multi-resolution central Hermite WENO (C-HWENO) schemes for solving hyperbolic conservation laws is proposed. Formulated in a central finite volume framework on staggered meshes, the methods adopt the multi-resolution HWENO reconstructions (Li et al. in J Comput Phys 446:110653, 2021; Li et al. in Commun Comput...
Article
In this paper, a new finite difference unequal-sized Hermite weighted essentially non-oscillatory (US-HWENO) scheme is proposed for solving multi-dimensional Navier-Stokes equations on structured meshes, which could achieve sixth-order accuracy in one dimension and fifth-order accuracy in two and three dimensions, respectively. The high-order spati...
Preprint
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In this paper, we propose a class of adaptive multiresolution (also called adaptive sparse grid) ultra-weak discontinuous Galerkin (UWDG) methods for solving some nonlinear dispersive wave equations including the Korteweg-de Vries (KdV) equation and its two dimensional generalization, the Zakharov-Kuznetsov (ZK) equation. The UWDG formulation, whic...
Article
Full-text available
The Hamilton-Jacobi (HJ) equations arise in optimal control and many other applications. Oftentimes, such equations are posed in high dimensions, and this presents great numerical challenges. In this paper, we propose an adaptive sparse grid (also called adaptive multiresolution) local discontinuous Galerkin (DG) method for solving Hamilton-Jacobi...
Article
This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The sparse grid method is a popular technique for high dimensional problems, and the associated collocation method has been well studied in the literature. The contribution of this work is the introduction of a systematic framework for col...
Article
Full-text available
This paper develops a high-order adaptive scheme for solving nonlinear Schrödinger equations. The solutions to such equations often exhibit solitary wave and local structures, which make adaptivity essential in improving the simulation efficiency. Our scheme uses the ultra-weak discontinuous Galerkin (DG) formulation and belongs to the framework of...
Article
Full-text available
In this paper, we propose a class of adaptive multiresolution (also called adaptive sparse grid) discontinuous Galerkin (DG) methods for simulating scalar wave equations in second order form in space. The two key ingredients of the schemes include an interior penalty DG formulation in the adaptive function space and two classes of multiwavelets for...
Preprint
Full-text available
This paper develops a high order adaptive scheme for solving nonlinear Schrodinger equations. The solutions to such equations often exhibit solitary wave and local structures, which makes adaptivity essential in improving the simulation efficiency. Our scheme uses the ultra-weak discontinuous Galerkin (DG) formulation and belongs to the framework o...
Preprint
Full-text available
We are interested in numerically solving the Hamilton-Jacobi (HJ) equations, which arise in optimal control and many other applications. Oftentimes, such equations are posed in high dimensions, and this poses great numerical challenges. This work proposes a class of adaptive sparse grid (also called adaptive multiresolution) local discontinuous Gal...
Preprint
Full-text available
In this paper, we propose a class of adaptive multiresolution (also called adaptive sparse grid) discontinuous Galerkin (DG) methods for simulating scalar wave equations in second order form in space. The two key ingredients of the schemes include an interior penalty DG formulation in the adaptive function space and two classes of multiwavelets for...
Preprint
This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The sparse grid method is a popular technique for high dimensional problems, and the associated collocation method has been well studied in the literature. The contribution of this work is the introduction of a systematic framework for col...
Article
Full-text available
In this paper, we develop sparse grid discontinuous Galerkin (DG) schemes for the Vlasov-Maxwell (VM) equations. The VM system is a fundamental kinetic model in plasma physics, and its numerical computations are quite demanding, due to its intrinsic high-dimensionality and the need to retain many properties of the physical solutions. To break the c...
Preprint
Full-text available
In this paper, we develop sparse grid central discontinuous Galerkin (CDG) scheme for linear hyperbolic systems with variable coefficients in high dimensions. The scheme combines the CDG framework with the sparse grid approach, with the aim of breaking the curse of dimensionality. A new hierarchical representation of piecewise polynomials on the du...
Preprint
Full-text available
In this paper, we develop sparse grid discontinuous Galerkin (DG) schemes for the Vlasov-Maxwell (VM) equations. The VM system is a fundamental kinetic model in plasma physics, and its numerical computations are quite demanding, due to its intrinsic high-dimensionality and the need to retain many properties of the physical solutions. To break the c...
Article
Full-text available
In this paper, a class of high-order central Hermite WENO (HWENO) schemes based on finite volume framework and staggered meshes is proposed for directly solving one- and two-dimensional Hamilton-Jacobi (HJ) equations. The methods involve the Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time. This w...
Article
Full-text available
In this paper, a class of high-order central finite volume schemes is proposed for solving one- and two-dimensional hyperbolic conservation laws. Formulated on staggered meshes, the methods involve Hermite WENO (HWENO) spatial reconstructions, and Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time....
Article
Full-text available
In this paper, we propose a class of high-order schemes for solving one- and two-dimensional hyperbolic conservation laws. The methods are formulated in a central finite volume framework on staggered meshes, and they involve Hermite WENO (HWENO) reconstructions in space, and Lax–Wendroff type discretizations or the natural continuous extension of R...

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