Shengtai Li

Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution

An adjoint sensitivity method is presented for parameter-dependent differential-algebraic equation systems (DAEs). The adjoint system is derived, along with conditions for its consistent initialization, for DAEs of index up to two (Hessenberg). For stable linear DAEs, stability of the adjoint system (for semi-explicit DAEs) or of an augmented adjoint system (for fully implicit DAEs) is shown. In addition, it is shown for these systems that numerical stability is maintained for the adjoint system or for the augmented adjoint system.

Software and algorithms for sensitivity analysis of large-scale differential algebraic systems

Sensitivity analysis for DAE systems is important in many engineering and scientific applications. The information contained in the sensitivity trajectories is useful for parameter estimation, optimization, model reduction and experimental design. In this paper we present algorithms and software for sensitivity analysis of large-scale DAE systems of index up to two. The new software provides for consistent initialization of the solutions and the sensitivities, interfaces seamlessly with automatic differentiation for the accurate evaluation of the sensitivity equations, and is capable via MPI of exploiting the natural parallelism of sensitivity analysis as well as providing an efficient solution in sequential computations.

Sensitivity analysis of differential-algebraic equations: A comparison of methods on a special problem ✩

We compare several methods for sensitivity analysis of differential‐algebraic equations (DAEs). Computational complexity, efficiency and numerical conditioning issues are discussed. Numerical results for a chemical kinetics problem arising in model reduction are presented.

Sensitivity analysis of differential-algebraic equations and partial differential equations

Sensitivity analysis generates essential information for model development, design optimization, parameter estimation, optimal control, model reduction and experimental design. In this paper we describe the forward and adjoint methods for sensitivity analysis, and outline some of our recent work on theory, algorithms and software for sensitivity analysis of differential-algebraic equation (DAE) and time-dependent partial differential equation (PDE) systems.

Adjoint sensitivity analysis for differential-algebraic equations: algorithms and software☆

An efficient numerical method for sensitivity computation of large-scale differential-algebraic systems is developed based on the adjoint method. Issues that are critical for the implementation are addressed. Complexity analysis and numerical results demonstrate that the adjoint sensitivity method is advantageous over the forward sensitivity method for applications with a large number of sensitivity parameters and few objective functions.

Stability of Moving Mesh Systems of Partial Differential Equations

Moving mesh methods based on the equidistribution principle (EP) are studied from the viewpoint of stability of the moving mesh system of differential equations. For fine spatial grids, the moving mesh system inherits the stability of the original discretized partial differential equation (PDE). Unfortunately, for some PDEs the moving mesh methods require so many spatial grid points that they no longer appear to be practical. Failures and successes of the moving mesh method applied to three reaction-diffusion problems are explained via an analysis of the stability and accuracy of the moving mesh PDE.

High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method

This paper extends the central finite-volume schemes of Liu et al. Y. Liu, C.-W. Shu, E. Tadmor, M. Zhang, Non-oscillatory hierarchical reconstruction for central and finite-volume schemes, Commun. Comput. Phys. 2 (2007) 933-963] on overlapping cells to the magneto-hydrodynamic (MHD) equations. In particular, we propose a high order divergence-free reconstruction for the magnetic field that uses the face-centered values. We also advance the magnetic field with a high order constrained transport (CT) scheme to preserve the divergence-free condition to machine round-off error. The overlapping cells are natural to be used to calculate the electric field flux without an averaging procedure. We have developed a third-order scheme which is verified by the numerical experiments. Other higher order schemes can be constructed accordingly. Our central constrained transport schemes do not need characteristic decomposition, and are easy to code and combine with un-split discretization of the source and parabolic terms. The overlapping cell representation of the solution is also used to develop more compact reconstruction and less dissipative schemes. The high resolution is achieved by non-oscillatory hierarchical reconstruction, which does not require characteristic decomposition either. The numerical comparisons show that the central schemes with non-CT perform as well as with CT for most of problems. Numerical examples are given to demonstrate efficacy of the new schemes.

An Adaptive Moving Mesh Method with Static Rezoning for Partial Differential Equations

Abstract Adaptive mesh methods are valuable tools in improving the accuracy and efficiency of the numerical solution of evolutionary systems of partial differential equations. If the mesh moves to track fronts and large gradients in the solution, then larger time steps can be taken than if it were to remain stationary. We derive explicit differential equations for moving the mesh so that the time variation of the solution at the mesh points is minimized. Moving the mesh based on this approach allows for larger time steps but does not guarantee that the solution is well resolved in space. We maintain spatial accuracy when there are new emerging layers or wave fronts by adaptively rezoning the mesh points to equidistribute an error estimate. When using a multistep integration method, the past solution values are also interpolated so that the same multistep method can be used after rezoning. The resulting algorithm has very few problem-dependent numerical parameters and is appropriate for a large class of one-dimensional partial differential equations. We illustrate the performance of the algorithm by examples and demonstrate that the proposed algorithm is efficient and accurate when compared with other adaptive mesh strategies.

Lie-Poisson integration for rigid body dynamics☆

Abstract In this paper, the splitting midpoint rule is presented and proved to be the Lie-Poisson integrators to the rigid body systems. Further discussions are also given. Numerical experiments show that this method has well properties comparing with the Runge Kutta method and ordinary midpoint rule.

A fourth-order divergence-free method for MHD flows

This paper extends our previous third-order method [S. Li, High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method, J. Comput. Phys. 227 (2008) 7368-7393] to the fourth-order. Central finite-volume schemes on overlapping grid are used for both the volume-averaged variables and the face-averaged magnetic field. The magnetic field at the cell boundaries falls within the dual grid and is naturally continuous so that our method eliminates the instability triggered by the discontinuity in the normal component of the magnetic field. Our fourth-order scheme has much smaller numerical dissipation than the third-order scheme. The divergence-free condition of the magnetic field is preserved by our fourth-order divergence-free reconstruction and the constrained transport method. Numerical examples show that the divergence-free condition is essential to the accuracy of the method when a limiter is used in the reconstruction. The high-order, low-dissipation, and divergence-free properties of this method make it an ideal tool for direct magneto-hydrodynamic turbulence simulations.