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Abstract The weighted essentially nonoscillatory (WENO) schemes, based on the successful essentially nonoscillatory (ENO) schemes with additional advantages, are a popular class of high-order accurate numerical methods for hyperbolic... more
Abstract The weighted essentially nonoscillatory (WENO) schemes, based on the successful essentially nonoscillatory (ENO) schemes with additional advantages, are a popular class of high-order accurate numerical methods for hyperbolic partial differential equations (PDEs) and other convection-dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high-order formal accuracy in smooth regions while maintaining stable, nonoscillatory and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution structures. In this chapter, we review the basic formulation of ENO and WENO schemes, outline the main ideas in constructing the schemes and discuss several of recent developments in using the schemes to solve hyperbolic type PDE problems.
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The interaction between an oblique shock wave and a pair of parallel vortices is simulated systematically through solving the two-dimensional, unsteady compressible Navier-Stokes equations using a fifth order weighted essentially... more
The interaction between an oblique shock wave and a pair of parallel vortices is simulated systematically through solving the two-dimensional, unsteady compressible Navier-Stokes equations using a fifth order weighted essentially nonoscillatory finite difference scheme. The main purpose of this study is to characterize the flow structure and the mechanism of sound generation in the interaction between an oblique shock wave and a pair of vortices. We study two typical shock waves of Mach number Ms=1.2 and Ms=1.05, which correspond to two typical shock structures of Mach reflection and regular reflection, respectively, in the problem of shock-vortex interaction. The effects of the strength of the vortices and the geometry parameters are investigated. In addition, we have also considered both cases of passing and colliding vortex pairs. The interaction is classified into four types for the passing case and seven types for the colliding case according to different patterns of the shock ...
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Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs). As other types of fast sweeping... more
Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs). As other types of fast sweeping schemes, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order. The resulting iterative schemes have fast convergence rate to steady state solutions. Moreover, an advantage of fixedpoint fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve inverse operation of any nonlinear local system. Hence they are robust and flexible, and have been combined with high order accurate weighted essentially non-oscillatory (WENO) schemes to solve various hyperbolic PDEs in the literature. For multidimensional nonlinear problems, high order fixed-point fast sweeping WE...
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A quantitative study is carried out in this paper to investigate the size of numerical viscosities and the resolution power of high-order weighted essentially nonoscillatory (WENO) schemes for solving one- and two-dimensional... more
A quantitative study is carried out in this paper to investigate the size of numerical viscosities and the resolution power of high-order weighted essentially nonoscillatory (WENO) schemes for solving one- and two-dimensional Navier-Stokes equations for compressible gas dynamics with high Reynolds numbers. A one-dimensional shock tube problem, a one-dimensional example with parameters motivated by supernova and laser experiments, and a two-dimensional Rayleigh-Taylor instability problem are used as numerical test problems. For the two-dimensional Rayleigh-Taylor instability problem, or similar problems with small-scale structures, the details of the small structures are determined by the physical viscosity (therefore, the Reynolds number) in the Navier-Stokes equations. Thus, to obtain faithful resolution to these small-scale structures, the numerical viscosity inherent in the scheme must be small enough so that the physical viscosity dominates. A careful mesh refinement study is pe...
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Research Interests: Engineering, Mathematics, Partial Differential Equations, Computer Science, Computational Physics, and 15 morePattern Formation, Finite element method, Morphogenesis, Mathematical Sciences, Diffusion, Mathematical Models, Discretization, PARTIAL DIFFERENTIAL EQUATION, Mathematical Model, Operator, High Dimensionality, Calculation, Discontinuous Galerkin, Embryo Development, and Factorization Method
Research Interests: Mathematics, Applied Mathematics, Computer Science, Computational Complexity, Discontinuous Galerkin Methods, and 10 moreFinite Element, Information Flow, Convergence Rate, First-Order Logic, Second Order, Finite Difference, Numerical Analysis and Computational Mathematics, High Order Accuracy, Discontinuous Galerkin, and Higher order
Contributed in honor of Fred Wan on the occasion of his 70th birthday Abstract. In a previous study [21], a class of efficient semi-implicit schemes was developed for stiff reaction-diffusion systems. This method which treats linear... more
Contributed in honor of Fred Wan on the occasion of his 70th birthday Abstract. In a previous study [21], a class of efficient semi-implicit schemes was developed for stiff reaction-diffusion systems. This method which treats linear diffusion terms exactly and nonlinear reaction terms implicitly has ex-cellent stability properties, and its second-order version, with a name IIF2, is linearly unconditionally stable. In this paper, we present another linearly un-conditionally stable method that approximates both diffusions and reactions implicitly using a second order Crank-Nicholson scheme. The nonlinear system resulted from the implicit approximation at each time step is solved using a multi-grid method. We compare this method (CN-MG) with IIF2 for their ac-curacy and efficiency. Numerical simulations demonstrate that both methods are accurate and robust with convergence using even very large size of time steps. IIF2 is found to be more accurate for systems with large diffusion while...
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In this paper, we construct a second order fast sweeping method with a discontinuous Galerkin
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Abstract. Skeletal patterning in the vertebrate limb, i.e., the spatiotemporal regulation of cartilage differentiation (chondrogenesis) during embryogenesis and regeneration, is one of the best studied examples of a multicellular... more
Abstract. Skeletal patterning in the vertebrate limb, i.e., the spatiotemporal regulation of cartilage differentiation (chondrogenesis) during embryogenesis and regeneration, is one of the best studied examples of a multicellular developmental process. Recently [Alber et al., The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb, Bulletin of Mathematical Biology, 2008, v70, pp. 460-483], a simplified two-equation reaction-diffusion system was developed to describe the interaction of two of the key morphogens: the activator and an activator-dependent inhibitor of precartilage condensation formation. A discontinuous Galerkin (DG) finite element method was applied to solve this nonlinear system on complex domains to study the effects of domain geometry on the pattern generated [Zhu et al., Application of Discontinuous Galerkin Methods for reaction-diffusion systems in developmental biology, Journal of Scientific Computing, 2009, v40, pp. 391-418]. In t...
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Abstract. The reaction-diffusion system modeling the dorsal-ventral pattern-ing during the zebrafish embryo development, developed in [Y.-T. Zhang, A.D. Lander, Q. Nie, Journal of Theoretical Biology, 248 (2007), 579–589] has mul-tiple... more
Abstract. The reaction-diffusion system modeling the dorsal-ventral pattern-ing during the zebrafish embryo development, developed in [Y.-T. Zhang, A.D. Lander, Q. Nie, Journal of Theoretical Biology, 248 (2007), 579–589] has mul-tiple steady state solutions. In this paper, we describe the computation of seven steady state solutions found by discretizing the boundary value problem using a finite difference scheme and solving the resulting polynomial system using algorithms from numerical algebraic geometry. The stability of each of these steady state solutions is studied by mathematical analysis and numeri-cal simulations via a time marching approach. The results of this paper show that three of the seven steady state solutions are stable and the location of the organizer of a zebrafish embryo determines which stable steady state pattern the multi-stability system converges to. Numerical simulations also show that the system is robust with respect to the change of the organizer size...
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We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R there exists a radially symmetric stationary solution with free... more
We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R there exists a radially symmetric stationary solution with free boundary r = R and interior boundary r = ρ.The system depends on a positive parameter μ, and for a sequence of values μ2 < μ3 < · · · there also exist branches of symmetric breaking stationary solutions, which bifurcate from these values. We use a homotopy method on the polynomial system associated to the discretization of the free boundary problem to compute the nonradial symmetric solutions and discuss their linear stability.
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We extend the weighted essentially non-oscillatory (WENO) schemes on two dimensional triangular meshes developed in [7] to three dimensions, and construct a third order finite volume WENO scheme on three dimensional tetrahedral meshes. We... more
We extend the weighted essentially non-oscillatory (WENO) schemes on two dimensional triangular meshes developed in [7] to three dimensions, and construct a third order finite volume WENO scheme on three dimensional tetrahedral meshes. We use the Lax-Friedrichs monotone flux as building blocks, third order reconstructions made from combinations of linear polynomials which are constructed on diversified small stencils of a tetrahedral mesh, and non-linear weights using smoothness indicators based on the derivatives of these linear polynomials. Numerical examples are given to demonstrate stability and accuracy of the scheme. AMS subject classifications: 65M99
The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of... more
The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear ...
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A fifth order weighted essentially non-oscillatory (WENO) fast sweeping method is designed in this paper, extending the result of the third order WENO fast sweeping method in [18] and utilizing the two approaches of accurate inflow... more
A fifth order weighted essentially non-oscillatory (WENO) fast sweeping method is designed in this paper, extending the result of the third order WENO fast sweeping method in [18] and utilizing the two approaches of accurate inflow boundary condition treatment in [6], which allows the usage of Cartesian meshes regardless of the domain boundary shape. The resulting method is tested on a variety of problems to demonstrate its good performance and CPU time efficiency when compared with lower order fast sweeping methods. keywords: fast sweeping method, WENO scheme, boundary condition.
The fundamental problem of pattern formation for example, how to specify body axes, limbs, digits during development comes down to interpreting a common set of genetic instructions differently at different locations in space (Wolpert... more
The fundamental problem of pattern formation for example, how to specify body axes, limbs, digits during development comes down to interpreting a common set of genetic instructions differently at different locations in space (Wolpert 1969). (See Figure 1.1 for an illustration). In all multicellular animals, this process is orchestrated by morphogens, molecules that are produced at discrete sites and disperse to form inspection ration gradients. Such gradients establish patterns because cells are preprogrammed to do very different things at different morphogen concentrations. Each cell responds to morphogens by reading their concentrations, and interprets them through intracellular machineries. A morphogen system usually consists of a region of morphogen-responsive cells, a region of morphogen producing cells, and a set of boundary conditions (Lander 2007). The objective of morphogen-responsive cells is to generate an intracellular signal, the amount of which reflects the level of mo...
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Fixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast... more
Fixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. Different from other fast sweeping methods, fixed-point iterative sweeping methods have the advantages such as that they have explicit forms and donotinvolve inverse operation of nonlinear local systems. In principle, it can be applied in solving very general equations using any monotone numerical fluxes and high order approximations easily. In this paper, based on the recently developed fifth order WENO schemes which improve the convergence of the classical WENO schemes by removing slight post-...
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Fixed-point fast sweeping WENO methods are a class of efficient high order numerical methods to solve steady state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping... more
Fixed-point fast sweeping WENO methods are a class of efficient high order numerical methods to solve steady state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence they are easy to be applied to a general hyperbolic system. In order to deal with the difficulties associated with numerical boundary treatment when high order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifth order fixed-p...
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Available in film copy from University Microfilms International. Thesis (Ph. D.)--Brown University, 2003. Vita. Thesis advisor: Chi-Wang Shu. Includes bibliographical references (leaves 123-128).
Research Interests: Embryos and Data Format
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In this paper, we review the major mathematical and computational models of vertebrate limb development and their roles in accounting for different aspects of this process. The main aspects of limb development that have been modeled... more
In this paper, we review the major mathematical and computational models of vertebrate limb development and their roles in accounting for different aspects of this process. The main aspects of limb development that have been modeled include outgrowth and shaping of the limb bud, establishment of molecular gradients within the bud, and formation of the skeleton. These processes occur interdependently during development, although (as described in this review), there are various interpretations of the biological relationships among them. A wide range of mathematical and computational methods have been used to study these processes, including ordinary and partial differential equation systems, cellular automata and discrete, stochastic models, finite difference methods, finite element methods, the immersed boundary method, and various combinations of the above. Multiscale mathematical modeling and associated computational simulation have become integrated into the study of limb morphogenesis and pattern formation to an extent with few parallels in the field of developmental biology. These methods have contributed to the design and analysis of experiments employing microsurgical and genetic manipulations, evaluation of hypotheses for limb bud outgrowth, interpretation of the effects of natural mutations, and the formulation of scenarios for the origination and evolution of the limb skeleton.