In this paper,we apply local discontinuous Galerkin(LDG)methods for pattern formation dynamical model in polymerizing actin focks.There are two main dificulties in designing effective numerical solvers.First of all,th...In this paper,we apply local discontinuous Galerkin(LDG)methods for pattern formation dynamical model in polymerizing actin focks.There are two main dificulties in designing effective numerical solvers.First of all,the density function is non-negative,and zero is an unstable equilibrium solution.Therefore,negative density values may yield blow-up solutions.To obtain positive numerical approximations,we apply the positivitypreserving(PP)techniques.Secondly,the model may contain stif source.The most commonly used time integration for the PP technique is the strong-stability-preserving Runge-Kutta method.However,for problems with stiff source,such time discretizations may require strictly limited time step sizes,leading to large computational cost.Moreover,the stiff source any trigger spurious filament polarization,leading to wrong numerical approximations on coarse meshes.In this paper,we combine the PP LDG methods with the semi-implicit Runge-Kutta methods.Numerical experiments demonstrate that the proposed method can yield accurate numerical approximations with relatively large time steps.展开更多
基金supported by the Natural Science Foundation of Shandong Province(ZR2021MA001)the Fundamental Research Funds for the Central Universities(20CX05011A)+1 种基金supported by National Natural Science Foundation of China Grant 11801569supported by NSF grant DMS-1818467 and Simons Foundation 961585.
文摘In this paper,we apply local discontinuous Galerkin(LDG)methods for pattern formation dynamical model in polymerizing actin focks.There are two main dificulties in designing effective numerical solvers.First of all,the density function is non-negative,and zero is an unstable equilibrium solution.Therefore,negative density values may yield blow-up solutions.To obtain positive numerical approximations,we apply the positivitypreserving(PP)techniques.Secondly,the model may contain stif source.The most commonly used time integration for the PP technique is the strong-stability-preserving Runge-Kutta method.However,for problems with stiff source,such time discretizations may require strictly limited time step sizes,leading to large computational cost.Moreover,the stiff source any trigger spurious filament polarization,leading to wrong numerical approximations on coarse meshes.In this paper,we combine the PP LDG methods with the semi-implicit Runge-Kutta methods.Numerical experiments demonstrate that the proposed method can yield accurate numerical approximations with relatively large time steps.
