Saul’yev and Group Explicit Methods

The Saul’yev methods for parabolic equations are implicit in form, but can be solved explicitly and are therefore interesting in connection with non-linear problems. Abdullah’s Group Explicit methods are parallel in nature and therefore interesting when using parallel computers. The  main objective of this paper is to study the accuracy of these methods. Using global error estimation we show that for all these methods the time step must be bounded by the square of the space step size to ensure a global error which can be estimated. As a curiosity we show that the two original Saul’yev methods in fact solve two different differential equations. &nbsp

Numerical Solution of Parabolic Equations

These lecture notes are designed for a one-semester course on finite-difference methods for parabolic equations. These equations which traditionally are used for describing diffusion and heat-conduction problems in Geology, Physics, and Chemistry have recently found applications in Finance Theory.Among the special features of this book can be mentioned the presentation of a practical approach to reliable estimates of the global error, including warning signals if the reliability is questionable. The technique is generally applicable for estimating the discretization error in numerical approximations which depend on a step size, such as numerical integration and solution of ordinary and partial differential equations. An integral part of the error estimation is the estimation of the order of the method and can thus satisfy the inquisitive mind: Is the order what we expect it to be from theopry ? and how do boundary value approximations affect the overall order of the method. Knowledge of a reliable order and error estimate enables us to determine (near-)optimal step sizes to meet a prescribed error tolerance, and possibly to extrapolate to get (higher order and) better accuracy at a minimal expense.Problems in two space dimensions are effectively handled using the Alternating Direction Implicit (ADI) technique. We present a systematic way of incorporating inhomogeneous terms and derivative boundary conditions in ADI methods as well as mixed derivative terms

Five Ways of Reducing the Crank-Nicolson Oscillations

Crank-Nicolson is a popular method for solving parabolic equations because it is unconditionally stable and second order accurate. One drawback of CN is that it responds to jump discontinuities in the initial conditions with oscillations which are weakly damped and therefore may persist for a long time. We present a selection of methods to reduce the amplitude of these oscillations

On the stability of ADI methods

When solving parabolic equations in two space dimensions implicit methods are preferred to the explicit method because of their better stability properties. Straightforward implementation of implicit methods require time-consuming solution of large systems of linear equations, and ADI methods are preferred instead. We expect the ADI methods to inherit the stability properties of the implicit methods they are derived from, and we demonstrate that this is partly true. The Douglas-Rachford and Peaceman-Rachford methods are absolutely stable in the sense that their growth factors are ≤ 1 in absolute value. Near jump discontinuities, however, there are differences w.r.t. how the ADI methods react to the situation: do they produce oscillations and how effectively do they damp them. We demonstrate the behaviour on two simple examples

Searching for a Space for Conversation: A study on how environment affects the articulation of the art experience among opera audiences at The Royal Danish Theatre

“Tell me about your experience in the theatre today”. Even though theatres (with this question) may seek honest, personal, and in depth answers from the audience, this question often results in superficial responds focused on what the audience suspect the theatre wants to hear. It can thus be difficult to get personal and detailed knowledge about the audiences’ experience. In a time, where theatres with different means (co-creation, participation etc.) try to keep audience loyal and engaged, this knowledge is important. In our project, we explore how different situations, locations, questions etc. affect conversations about art experiences and in this article we describe our “search for a space for conversation”. We explore how the space affects the conversation, and how the setting can emphasize certain elements. What happens to the conversation if we sit around an ordinary meeting table, if we walk and talk outside the art institution or if we talk inside the auditorium, where we had the original art experience? This explorative study is part of the project “A Suitcase of Methods”, which is housed by The Royal Danish Theatre and financially supported by The Bikuben Foundation

Effect of Combination Therapy with a Calcium Channel Blocker and an Angiotensin-Converting Enzyme Inhibitor on Renal Hypertrophy and Urinary Albumin Excretion in Diabetic Rats

The objective of this study was to compare the effect of an angiotensin-converting enzyme (ACE) inhibitor and a calcium channel blocker on the development of renal changes in diabetic rats. Diabetes was induced by an intravenous injection of streptozotocin in normotensive Wistar rats. Treatment was commenced immediately in 1 set of rats with 4 treatment arms: nitrendipine (250 mg/kg fodder), enalapril (35 mg/L drinking water), both treatments in combination, or placebo. Treatment was continued for 9 weeks. Another set of rats was left with untreated diabetes for 3 months followed by 7 weeks treatment as above. When starting treatment right after induction of diabetes, nitrendipine significantly reduced urinary albumin excretion (UAE) to the nondiabetic level (P < .05) without reducing blood pressure (BP), whereas enalapril failed to significantly reduce UAE despite a reduction in BP. Combining the two treatments showed no further reduction in UAE compared to monotherapy with nitrendipine, despite a lower BP. When leaving diabetic rats untreated for 3 months, only the coadministration of nitrendipine and enalapril showed a significant reduction in UAE compared to monotherapy and placebo treatment, but showed no significant effect on BP

Hvorfor er matematik så populært i 7. a? En undersøgelse af elevers grunde til at værdsætte færdighedsorienteret matematikundervisning

En af barriererne mod forandring af den færdighedsorienterede matematikundervisning der præger mange folkeskoleklasser, er at den fungerer godt i en skolekontekst. Sådan er det umiddelbare billede også i den 7.-klasse vi har fulgt gennem observationer og interviews. Her er matematik det højst værdsatte fag blandt elever med meget forskellige baggrunde. I vores analyse af elevinterviewene stiller vi skarpt på elevernes grunde til at kunne lide matematik. Analysen peger på at matematikundervis-ningens fortrin set fra elevperspektiv skal findes i en tydelig elevrolle, oplevelsen af læringsprogression og forestillinger om det lærtes betydning for fremtiden