Stiffly accurate Runge-Kutta methods for stiff stochastic differential equations

In this paper we discuss implicit methods based on stiffly accurate Runge-Kutta methods and splitting techniques for solving Stratonovich stochastic differential equations (SDEs). Two splitting techniques: the balanced splitting technique and the deterministic splitting technique, are used in this paper. We construct a two-stage implicit Runge-Kutta method with strong order 1.0 which is corrected twice and no update is needed. The stability properties and numerical results show that this approach is suitable for solving stiff SDEs. (C) 2001 Elsevier Science B.V. All rights reserved.

Predictor-corrector methods of Runge-Kutta type for stochastic differential equations

In this paper we construct predictor-corrector (PC) methods based on the trivial predictor and stochastic implicit Runge-Kutta (RK) correctors for solving stochastic differential equations. Using the colored rooted tree theory and stochastic B-series, the order condition theorem is derived for constructing stochastic RK methods based on PC implementations. We also present detailed order conditions of the PC methods using stochastic implicit RK correctors with strong global order 1.0 and 1.5. A two-stage implicit RK method with strong global order 1.0 and a four-stage implicit RK method with strong global order 1.5 used as the correctors are constructed in this paper. The mean-square stability properties and numerical results of the PC methods based on these two implicit RK correctors are reported.

A four-stage index 2 Diagonally Implicit Runge-Kutta method

High index Differential Algebraic Equations (DAEs) force standard numerical methods to lower order. Implicit Runge-Kutta methods such as RADAU5 handle high index problems but their fully implicit structure creates significant overhead costs for large problems. Singly Diagonally Implicit Runge-Kutta (SDIRK) methods offer lower costs for integration. This paper derives a four-stage, index 2 Explicit Singly Diagonally Implicit Runge-Kutta (ESDIRK) method. By introducing an explicit first stage, the method achieves second order stage calculations. After deriving and solving appropriate order conditions., numerical examples are used to test the proposed method using fixed and variable step size implementations. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.

Implicit stochastic Runge-Kutta methods for stochastic differential equations

In this paper we construct implicit stochastic Runge-Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods.

On functionally-fitted Runge-Kutta methods

Functionally-fitted methods are generalizations of collocation techniques to integrate an equation exactly if its solution is a linear combination of a chosen set of basis functions. When these basis functions are chosen as the power functions, we recover classical algebraic collocation methods. This paper shows that functionally-fitted methods can be derived with less restrictive conditions than previously stated in the literature, and that other related results can be derived in a much more elegant way. The novelty in our approach is to fully retain the collocation framework without reverting back into derivations based on cumbersome Taylor series expansions.

Analysis of trigonometric implicit Runge-Kutta methods

Using generalized collocation techniques based on fitting functions that are trigonometric (rather than algebraic as in classical integrators), we develop a new class of multistage, one-step, variable stepsize, and variable coefficients implicit Runge-Kutta methods to solve oscillatory ODE problems. The coefficients of the methods are functions of the frequency and the stepsize. We refer to this class as trigonometric implicit Runge-Kutta (TIRK) methods. They integrate an equation exactly if its solution is a trigonometric polynomial with a known frequency. We characterize the order and A-stability of the methods and establish results similar to that of classical algebraic collocation RK methods. (c) 2006 Elsevier B.V. All rights reserved.

High selectivity microporous silica membranes for lactic acid dehydration

Lactic acid (LA) has significant market potential for many industries including food, cosmetics, pharmaceuticals, medical and biodegradable materials. Production of LA usually begins with the fermentation of glucose but subsequent stages for the enrichment of lactic acid are complex and energy intensive and could be minimised using water selective membrane technology. In this work, we trialled a highly selective hydrostable carbonised template molecular sieve silica (CTMSS) membrane for the dehydration of a 15 vol% aqueous lactic acid solution with 0.1 vol% glucose. CTMSS membrane films were developed by dip-coating ceramic substrates with silica sols made using the acid catalysed sol-gel process. Permeation was performed by feeding LA/glucose solution to the membrane cell at 18°C in a standard pervaporation setup. The membrane showed selective transport of water from the aqueous feed to the permeate while glucose was not detected. CTMSS membrane permeate flux stabilised at 0.2 kg.m-2.hr-1 in 3.9 hours, and reduced LA to lower than 0.2 vol%. Flux through the CTMSS micropores was activated, displaying increased initial flux to 1.58 kg.m-2.hr-1 at 60°C. To enrich a 1 l.min-1 stream to 85% LA in a single stage, a minimum membrane area of 324 m2 would be required at 18°C. Increased operating temperature to 80°C significantly reduced this area to 24 m2 but LA levels in the permeate stream increased to 0.5 vol%. The highly selective CTMSS membrane technology is an ideal candidate for LA purification. CTMSS membrane systems operate stably in aqueous systems leading to potential cost reductions in LA processing for future markets.

Fault-tolerant quantum computation with cluster states

The one-way quantum computing model introduced by Raussendorf and Briegel [Phys. Rev. Lett. 86, 5188 (2001)] shows that it is possible to quantum compute using only a fixed entangled resource known as a cluster state, and adaptive single-qubit measurements. This model is the basis for several practical proposals for quantum computation, including a promising proposal for optical quantum computation based on cluster states [M. A. Nielsen, Phys. Rev. Lett. (to be published), quant-ph/0402005]. A significant open question is whether such proposals are scalable in the presence of physically realistic noise. In this paper we prove two threshold theorems which show that scalable fault-tolerant quantum computation may be achieved in implementations based on cluster states, provided the noise in the implementations is below some constant threshold value. Our first threshold theorem applies to a class of implementations in which entangling gates are applied deterministically, but with a small amount of noise. We expect this threshold to be applicable in a wide variety of physical systems. Our second threshold theorem is specifically adapted to proposals such as the optical cluster-state proposal, in which nondeterministic entangling gates are used. A critical technical component of our proofs is two powerful theorems which relate the properties of noisy unitary operations restricted to act on a subspace of state space to extensions of those operations acting on the entire state space. We expect these theorems to have a variety of applications in other areas of quantum-information science.

Entanglement in a simple quantum phase transition

What entanglement is present in naturally occurring physical systems at thermal equilibrium? Most such systems are intractable and it is desirable to study simple but realistic systems that can be solved. An example of such a system is the one-dimensional infinite-lattice anisotropic XY model. This model is exactly solvable using the Jordan-Wigner transform, and it is possible to calculate the two-site reduced density matrix for all pairs of sites. Using the two-site density matrix, the entanglement of formation between any two sites is calculated for all parameter values and temperatures. We also study the entanglement in the transverse Ising model, a special case of the XY model, which exhibits a quantum phase transition. It is found that the next-nearest-neighbor entanglement (though not the nearest-neighbor entanglement) is a maximum at the critical point. Furthermore, we show that the critical point in the transverse Ising model corresponds to a transition in the behavior of the entanglement between a single site and the remainder of the lattice.

Theory manual to OctVCE – a cartesian cell CFD code with special application to blast wave problems, Department of Mechanical Engineering Report 2007/12

OctVCE is a cartesian cell CFD code produced especially for numerical simulations of shock and blast wave interactions with complex geometries, in particular, from explosions. Virtual Cell Embedding (VCE) was chosen as its cartesian cell kernel for its simplicity and sufficiency for practical engineering design problems. The code uses a finite-volume formulation of the unsteady Euler equations with a second order explicit Runge-Kutta Godonov (MUSCL) scheme. Gradients are calculated using a least-squares method with a minmod limiter. Flux solvers used are AUSM, AUSMDV and EFM. No fluid-structure coupling or chemical reactions are allowed, but gas models can be perfect gas and JWL or JWLB for the explosive products. This report also describes the code’s ‘octree’ mesh adaptive capability and point-inclusion query procedures for the VCE geometry engine. Finally, some space will also be devoted to describing code parallelization using the shared-memory OpenMP paradigm. The user manual to the code is to be found in the companion report 2007/13.

Modeling ex vivo hematopoiesis using chemical engineering metaphors

Ex vivo hematopoiesis is increasingly used for clinical applications. Models of ex vivo hematopoiesis are required to better understand the complex dynamics and to optimize hematopoietic culture processes. A general mathematical modeling framework is developed which uses traditional chemical engineering metaphors to describe the complex hematopoietic dynamics. Tanks and tubular reactors are used to describe the (pseudo-) stochastic and deterministic elements of hematopoiesis, respectively. Cells at any point in the differentiation process can belong to either an immobilized, inert phase (quiescent cells) or a mobile, active phase (cycling cells). The model describes five processes: (1) flow (differentiation), (2) autocatalytic formation (growth),(3) degradation (death), (4) phase transition from immobilized to mobile phase (quiescent to cycling transition), and (5) phase transition from mobile to immobilized phase (cycling to quiescent transition). The modeling framework is illustrated with an example concerning the effect of TGF-beta 1 on erythropoiesis. (C) 1998 Published by Elsevier Science Ltd. All rights reserved.

Structure-activity studies of conantokins as human N-methyl-D-aspartate receptor modulators

The activities of conantokin-G (con-G), conantokin-T (con-T), and several novel analogues have been studied using polyamine enhancement of [H-3]MK-801 binding to human glutamate-N-methyl-D-aspartate (NMDA) receptors, and their structures have been examined using CD and H-1 NMR spectroscopy. The potencies of con-G[A7], con-G, and con-T as noncompetitive inhibitors of spermine-enhanced [H-3]MK-801 binding to NMDA receptor obtained from human brain tissue are similar to those obtained using rat brain tissue. The secondary structure and activity of con-G are found to be highly sensitive to amino acid substitution and modification. NMR chemical shift data indicate that con-G, con-G[D8,D17], and con-G[A7] have similar conformations in the presence of Ca2+. This consists of a helix for residues 2-16, which is kinked in the vicinity of Gla10. This is confirmed by 3D structure calculations on con-G[A7]. Restraining this helix in a linear form (i.e., con-G[A7,E10-K13]) results in a minor reduction in potency. Incorporation of a 7-10 salt-bridge replacement (con-G[K7-E10]) prevents helix formation in aqueous solution and produces a peptide with low potency. Peptides with the Leu5-Tyr5 substitution also have low potencies (con-G[Y5,A7] and con-G[Y5,K7]) indicating that Leu5 in con-G is important for full antagonist behavior. We have also shown that the Gla-Ala7 substitution increases potency, whereas the Gla-Lys7 substitution has no effect. Con-G and con-G[K7] both exhibit selectivity between NMDA subtypes from mid-frontal and superior temporal gyri, but not between sensorimotor and mid-frontal gyri. Asn8 and/or Asn17 appear to be important for the ability of con-G to function as an inhibitor of polyamine-stimulated [3H]MK-801 binding, but not in maintaining secondary structure. The presence of Ca2+ does not increase the potencies of con-G and con-T for NMDA receptors but does stabilize the helical structures of con-G, con-G[D8,D17], and, to a lesser extent, con-G[A7]. The NMR data support the existence of at least two independent Ca2+-chelating sites in con-G, one involving Gla7 and possibly Gla3 and the other likely to involve Gla10 and/or Gla14.