arXiv subjects

Cao Wen

Publications and source records attributed to Cao Wen.

3 recordsLinked to original sources

arXiv2026

Long-time stability of implicit-explicit Runge-Kutta methods for two-dimensional incompressible flows

High-order adaptive time-stepping algorithms are of significant practical value and theoretical interest for accelerating long-time fluid-flow simulations and resolving complex dynamical behaviors. While several high-order implicit-explicit schemes have been proposed in the literature, their long-time stability properties remain largely unexplored. We develop a family of long-time stable implicit-explicit Runge-Kutta (IERK) methods, up to fourth-order temporal accuracy, for the two-dimensional incompressible Navier-Stokes equations in vorticity-stream function formulation. By combining a convolution-type H\"{o}lder inequality with a damping-type multistage Gr\"{o}nwall inequality, we establish a unified analytical framework that proves long-time stability in both the $L^2$ and $H^1$ norms. A key component of the analysis is a mathematical-induction argument that ensures stage-wise boundedness of the vorticity in the $H^\delta$ norm for some $\delta>0$. To the best of our knowledge, this is the first work to establish large-time stability results for high-order IERK algorithms for the two-dimensional incompressible Navier-Stokes equations. Our IERK schemes employ stiffly accurate diagonally implicit Runge-Kutta approximations for the linear diffusive term together with explicit Runge-Kutta approximations for the nonlinear advection term. By exploiting the specific structure of the Navier-Stokes model, we derive a reduced set of order conditions-requiring only 5 and 11 conditions for the third- and fourth-order methods, respectively, in contrast to the classical 6 and 18-allowing the construction of a parameterized family of efficient schemes. These IERK methods are particularly well suited for adaptive time-stepping, as they permit significantly enlarged step sizes in actual computations.

math.NA↗

arXiv2024

Average energy dissipation rates of additive implicit-explicit Runge-Kutta methods for gradient flow problems

A unified theoretical framework is suggested to examine the energy dissipation properties at all stages of additive implicit-explicit Runge-Kutta (IERK) methods up to fourth-order accuracy for gradient flow problems. We construct some parameterized IERK methods by applying the so-called first same as last method, that is, the diagonally implicit Runge-Kutta method with the explicit first stage and stiffly-accurate assumption for the linear stiff term, and applying the explicit Runge-Kutta method for the nonlinear term. The main part of the novel framework is to construct the differential forms and the associated differentiation matrices of IERK methods by using the difference coefficients of method and the so-called discrete orthogonal convolution kernels. As the main result, we prove that an IERK method can preserve the original energy dissipation law unconditionally if the associated differentiation matrix is positive semi-definite. The recent indicator, namely average energy dissipation rate, is also adopted for these multi-stage methods to evaluate the overall energy dissipation rate of an IERK method such that one can choose proper parameters in some parameterized IERK methods. It is found that the selection of method parameters in the IERK methods is at least as important as the selection of different IERK methods. Extensive numerical experiments are also included to support our theory.

math.NA↗

arXiv2024

Original energy dissipation preserving corrections of integrating factor Runge-Kutta methods for gradient flow problems

Explicit integrating factor Runge-Kutta methods are attractive and popular in developing high-order maximum bound principle preserving time-stepping schemes for Allen-Cahn type gradient flows. However, they always suffer from the non-preservation of steady-state solution and original energy dissipation law. To overcome these disadvantages, some new integrating factor methods are developed by using two classes of difference correction, including the telescopic correction and nonlinear-term translation correction, enforcing the preservation of steady-state solution. Then the original energy dissipation properties of the new methods are examined by using the associated differential forms and the differentiation matrices. As applications, some new integrating factor Runge-Kutta methods up to third-order maintaining the original energy dissipation law are constructed by applying the difference correction strategies to some popular explicit integrating factor methods in the literature. Extensive numerical experiments are presented to support our theory and to demonstrate the improved performance of new methods.

math.NA↗