A partially penalised immersed finite element method for interface problems with discontinuous coefficients and non-homogeneous jump conditions based on unfitted meshes independent of the interface is proposed. The arising systems of linear equations have symmetric positive definite matrices which allows the use of fast solvers and existing codes. Optimal error estimates in an energy norm are derived. Numerical examples demonstrate the efficiency of the method.

We discuss spectral collocation methods based on Jacobi-Gauss-Lobatto points and Laguerre and Hermite collocation for unbounded domains. These methods are well conditioned, and some numerical experiments demonstrate quite high accuracy.

Minimisation of the total variation regularisation for linear operators under $ℓ_1$-constraints applied to image restoration is considered, and relationships between the Lagrange multiplier for a constrained model and the regularisation parameter in an unconstrained model are established. A constrained $ℓ_1$-problem reformulated as a separable convex problem is solved by the alternating direction method of multipliers that includes two sequences, converging to a restored image and the “optimal" regularisation parameter. This allows blurry images to be recovered, with a simultaneous estimation of the regularisation parameter. The noise level parameter is estimated, and numerical experiments illustrate the efficiency of the new approach.

A recent accelerated SOR-like method for generalised saddle point problems is discussed. Sufficient conditions for convergence are derived, and some numerical experiments illustrate its effectiveness.

Based on the generalised Arnoldi procedure, we develop an implicitly restarted generalised Arnoldi method for solving the large-scale polynomial eigenvalue problem. By combining implicit restarting with the refinement scheme, we present an implicitly restarted refined generalised Arnoldi (IRGAR) method. To avoid repeated converged eigenpairs in the later iteration, we develop a novel non-equivalence low-rank deflation technique and propose a deflated and implicitly restarted refined generalised Arnoldi method (DIRGAR). Some numerical experiments show that this DIRGAR method is efficient and robust.

The generalised perturbation $(n, N−n)$-fold Darboux transformation is used to derive new higher-order rogue wave and rational soliton solutions of the discrete complex mKdV equations. The structure of such waves and details of their evolution are investigated via numerical simulations, showing that the strong interaction yields weak oscillation and stability whereas the weak interaction is associated with strong oscillation and instability. A small noise has a weak influence on the wave propagation for the strong interaction, but substantially changes the wave behaviour in the weak interaction case.

American put options on a zero-coupon bond problem is reformulated as a linear complementarity problem of the option value and approximated by a nonlinear partial differential equation. The equation is solved by an exponential time differencing method combined with a barycentric Legendre interpolation and the Krylov projection algorithm. Numerical examples shows the stability and good accuracy of the method.

A modification to the Newton method for nonlinear eigenvalue problems is proposed and locally quadratic convergence of this algorithm is established. Numerical examples show the efficiency of the method and reduced computational cost.

Hyperbolic, rotation invariant moment systems are derived for a non-linear kinetic description of two-dimensional Vicsek swarming model. The systems also preserve mass conservation, and numerical experiments show that this approach captures the key features of the model such as shock waves, contact discontinuities, rarefaction waves, vortex formations. If the system order increases, the solutions of the moment systems converge to the solution of the corresponding kinetic equation.

The stability and convergence of a weak Galerkin finite element method for multi-term time-fractional diffusion equations with one-dimensional space variable are proved. Numerical experiments are consistent with theoretical analysis.

The Nyström method for double layer potential equations on simple closed piecewise smooth contours is stable if and only if certain operators associated with the opening angles of the corners are invertible. Numerical experiments show that there are opening angles which cause the instability of the method.