Advanced Time Stepping Methods for Large-Scale ODEs and PDEs
Time stepping methods are algorithms used to compute the numerical solution of ordinary differential equations as well as to evolve the solution of partial differential equations in time. Based on successful time integration, sensitivity analysis quantifies the relationship between changes in model parameters and changes in model output, which is crucial for model-constrained optimization, inverse problems and uncertainty quantification. ODE solvers with sensitivity analysis capabilities are thus highly desirable.
I have developedFATODE, the first publicly available general purpose library for solution of stiff and non- stiff ODEs that offers forward and discrete adjoint sensitivity analysis capabilities in the context of Runge-Kutta methods. High performance has been demonstrated on shallow water equations, CBM-4 chemical mechanism, multi-body vehicle dynamics and so on. Implicit-explicit (IMEX) time stepping methods can efficiently solve differential equations with both stiff and nonstiffcomponents. In this talk, I will introduce my research on new implicit-explicit methods of general linear type (IMEX-GLMs). I established an order conditions theory for high stage order partitioned GLMs that share the same abscissae, and show that no additional coupling order conditions are needed. Consequently, GLMs offer an excellent framework for the construction of multi-method integration algorithms.
I have proposed a family of IMEX schemes based on diagonally-implicit multi- stage integration methods and constructed practical schemes of orders up to five. The new schemes can avoid possible order reduction which can arise with Runge-Kutta schemes. This property is essential for the success of high order IMEX schemes. I have also developed a family of IMEX two-step Runge- Kutta methods with a stiffly accurate implicit part. Both families of IMEX-GLMs have been implemented into a high performance geophysical simulation software which uses discontinuous Galerkin methods for space discretization. Numerical results confirm the theoretical findings.