A non-semisimple matrix loop algebra is presented, and a class of zero curvature equations over this loop algebra is used to generate bi-integrable couplings. An illustrative example is made for the Dirac soliton hierarchy. Associated variational identities yield bi-Hamiltonian structures of the resulting bi-integrable couplings, such that the hierarchy of bi-integrable couplings possesses infinitely many commuting symmetries and conserved functionals.

A spectral-element method is developed to solve the scattering problem for time-harmonic sound waves due to an obstacle in an homogeneous compressible fluid. The method is based on a boundary perturbation technique coupled with an efficient spectral-element solver. Extensive numerical results are presented, in order to show the accuracy and stability of the method.

In this article, a fully discrete finite element approximation is investigated for constrained parabolic optimal control problems with time-dependent coefficients. The spatial discretisation invokes finite elements, and the time discretisation a nonstandard backward Euler method. On introducing some appropriate intermediate variables and noting properties of the $L^2$ projection and the elliptic projection, we derive the superconvergence for the control, the state and the adjoint state. Finally, we discuss some numerical experiments that illustrate our theoretical results.

In this article, we derive a new fourth-order finite difference formula based on off-step discretisation for the solution of two-dimensional nonlinear triharmonic partial differential equations on a 9-point compact stencil, where the values of $u$, $(∂^{2}u/∂n^2)$ and $(∂^{4}u/∂n^{4})$ are prescribed on the boundary. We introduce new ways to handle the boundary conditions, so there is no need to discretise the boundary conditions involving the partial derivatives. The Laplacian and biharmonic of the solution are obtained as a by-product of our approach, and we only need to solve a system of three equations. The new method is directly applicable to singular problems, and we do not require any fictitious points for computation. We compare its advantages and implementation with existing basic iterative methods, and numerical examples are considered to verify its fourth-order convergence rate.

We develop an unconditionally energy stable immersed boundary method, and apply it to simulate 2D vesicle dynamics. We adopt a semi-implicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the projection method to solve the fluid equations, the pressure is decoupled and we have a symmetric positive definite system that can be solved efficiently. The method can be shown to be unconditionally stable, in the sense that the total energy is decreasing. A resulting modification benefits from this improved numerical stability, as the time step size can be significantly increased (the severe time step restriction in an explicit boundary forcing scheme is avoided). As an application, we use our scheme to simulate vesicle dynamics in Navier-Stokes flow.