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We present a mass-conservative vertex-centred finite volume method for efficiently solving the mixed form of Richards’ equation in heterogeneous porous media. The spatial discretisation is particularly well-suited to heterogeneous media because it produces consistent flux approximations at quadrature points where material properties are continuous. Combined with the method of lines, the spatial discretisation gives a set of differential algebraic equations amenable to solution using higher-order implicit solvers. We investigate the solution of the mixed form using a Jacobian-free inexact Newton solver, which requires the solution of an extra variable for each node in the mesh compared to the pressure-head form. By exploiting the structure of the Jacobian for the mixed form, the size of the preconditioner is reduced to that for the pressure-head form, and there is minimal computational overhead for solving the mixed form. The proposed formulation is tested on two challenging test problems. The solutions from the new formulation offer conservation of mass at least one order of magnitude more accurate than a pressure head formulation, and the higher-order temporal integration significantly improves both the mass balance and computational efficiency of the solution.

We survey preconditioned iterative methods with the emphasis on solving large sparse systems such as arise by discretization of boundary value problems for partial differential equations. We discuss shortly various acceleration methods but the main emphasis is on efficient preconditioning techniques. Numerical simulations on practical problems have indicated that an efficient preconditioner is the most important part of an iterative algorithm. We report in particular on the state of the art of preconditioning methods for vectorizable and/or parallel computers.

In this paper, we construct and analyze a new dynamical low-rank integrator for second-order matrix differential equations. The method is based on a combination of the projector-splitting integrator introduced in Lubich and Oseledets (BIT 54(1):171–188, 2014. https://doi.org/10.1007/s10543-013-0454-0 ) and a Strang splitting. We also present a variant of the new integrator which is tailored to semilinear second-order problems.

The operator splitting method in combination with finite differences has been shown to be an efficient approach for pricing American options numerically. Here, the operator splitting formulation is extended to the radial basis function partition of unity method. An approach that has previously often been used together with radial basis function methods to deal with the free boundary arising in American option pricing is to solve a penalised version of the Black–Scholes equation. It is shown that the operator splitting technique outperforms the penalty approach when used with the radial basis function partition of unity method. Numerical experiments are performed for one, two and three underlying assets. The advantage of the operator splitting technique grows with the number of dimensions.

This article is concerned with geometric integrators which are linearization-preserving, i.e. numerical integrators which preserve the exact linearization at every fixed point of an arbitrary system of ODEs. For a canonical Hamiltonian system, we propose a new symplectic and self-adjoint B-series method which is also linearization-preserving. In a similar fashion, we show that it is possible to construct a self-adjoint and linearization-preserving B-series method for an arbitrary system of ODEs. Some numerical experiments on Hamiltonian ODEs are presented to test the behaviour of both proposed methods.

Equivalence of Hadamard matrices can be decided inO(log2 n) space, and hence in subexponential time. These resource bounds follow from the existence of small distinguishing sets.