In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-depe...In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.展开更多
In this paper,the second in a series,we improve the discretization of the higher spatial derivative terms in a spectral volume(SV)context.The motivation for the above comes from[J.Sci.Comput.,46(2),314–328],wherein t...In this paper,the second in a series,we improve the discretization of the higher spatial derivative terms in a spectral volume(SV)context.The motivation for the above comes from[J.Sci.Comput.,46(2),314–328],wherein the authors developed a variant of the LDG(Local Discontinuous Galerkin)flux discretization method.This variant(aptly named LDG2),not only displayed higher accuracy than the LDG approach,but also vastly reduced its unsymmetrical nature.In this paper,we adapt the LDG2 formulation for discretizing third derivative terms.A linear Fourier analysis was performed to compare the dispersion and the dissipation properties of the LDG2 and the LDG formulations.The results of the analysis showed that the LDG2 scheme(i)is stable for 2nd and 3rd orders and(ii)generates smaller dissipation and dispersion errors than the LDG formulation for all the orders.The 4th order LDG2 scheme is howevermildly unstable:as the real component of the principal eigen value briefly becomes positive.In order to circumvent the above,a weighted average of the LDG and the LDG2 fluxes was used as the final numerical flux.Even a weight of 1.5%for the LDG(i.e.,98.5%for the LDG2)was sufficient tomake the scheme stable.Thisweighted scheme is still predominantly LDG2 and hence generated smaller dissipation and dispersion errors than the LDG formulation.Numerical experiments are performed to validate the analysis.In general,the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries(KdV)type problems.展开更多
In this paper,we develop a formulation for solving equations containing higher spatial derivative terms in a spectral volume(SV)context;more specifically the emphasis is on handling equations containing third derivati...In this paper,we develop a formulation for solving equations containing higher spatial derivative terms in a spectral volume(SV)context;more specifically the emphasis is on handling equations containing third derivative terms.This formulation is based on the LDG(Local Discontinuous Galerkin)flux discretization method,originally employed for viscous equations containing second derivatives.A linear Fourier analysis was performed to study the dispersion and the dissipation properties of the new formulation.The Fourier analysis was utilized for two purposes:firstly to eliminate all the unstable SV partitions,secondly to obtain the optimal SV partition.Numerical experiments are performed to illustrate the capability of this formulation.Since this formulation is extremely local,it can be easily parallelized and a h-p adaptation is relatively straightforward to implement.In general,the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries(KdV)type problems.展开更多
文摘In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.
文摘In this paper,the second in a series,we improve the discretization of the higher spatial derivative terms in a spectral volume(SV)context.The motivation for the above comes from[J.Sci.Comput.,46(2),314–328],wherein the authors developed a variant of the LDG(Local Discontinuous Galerkin)flux discretization method.This variant(aptly named LDG2),not only displayed higher accuracy than the LDG approach,but also vastly reduced its unsymmetrical nature.In this paper,we adapt the LDG2 formulation for discretizing third derivative terms.A linear Fourier analysis was performed to compare the dispersion and the dissipation properties of the LDG2 and the LDG formulations.The results of the analysis showed that the LDG2 scheme(i)is stable for 2nd and 3rd orders and(ii)generates smaller dissipation and dispersion errors than the LDG formulation for all the orders.The 4th order LDG2 scheme is howevermildly unstable:as the real component of the principal eigen value briefly becomes positive.In order to circumvent the above,a weighted average of the LDG and the LDG2 fluxes was used as the final numerical flux.Even a weight of 1.5%for the LDG(i.e.,98.5%for the LDG2)was sufficient tomake the scheme stable.Thisweighted scheme is still predominantly LDG2 and hence generated smaller dissipation and dispersion errors than the LDG formulation.Numerical experiments are performed to validate the analysis.In general,the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries(KdV)type problems.
文摘In this paper,we develop a formulation for solving equations containing higher spatial derivative terms in a spectral volume(SV)context;more specifically the emphasis is on handling equations containing third derivative terms.This formulation is based on the LDG(Local Discontinuous Galerkin)flux discretization method,originally employed for viscous equations containing second derivatives.A linear Fourier analysis was performed to study the dispersion and the dissipation properties of the new formulation.The Fourier analysis was utilized for two purposes:firstly to eliminate all the unstable SV partitions,secondly to obtain the optimal SV partition.Numerical experiments are performed to illustrate the capability of this formulation.Since this formulation is extremely local,it can be easily parallelized and a h-p adaptation is relatively straightforward to implement.In general,the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries(KdV)type problems.
