The first main focus in the present project was to analyse the boundary treatment of the linearised Korteweg-de Vries equation. The second main focus was to derive a stable numerical solution using a high-order finite difference method. Since the model involved a third derivative in space, the numerical treatment of the boundaries was highly nontrivial. To aid the boundary treatment high-order accurate first and third derivative finite difference operators were employed. The boundaries are based on the summation-by-parts (SBP) framework, thereby guaranteeing linear stability. The boundary conditions were imposed using a penalty technique. A convergence study was performed where the derived numerical solution was compared with an analytical one. Fourth order accurate Runge-Kutta was used to time-integrate the numerical approximation. Measuring the rate of convergence, q, yielded q = 4 for 4th order accurate SBP-operators and q = 5.5 for 6th order accurate SBP-operators. Thus the convergence study proved the accuracy and stability of the numerical solution derived with the SBP-methodology.

No description available.

L’hypertrophie bénigne de la prostate (HBP) est une pathologie qui affecte 50% des hommes dès l’âge de 60 ans et qui conduit à des troubles de la miction. L’HBP se caractérise par une hypertrophie exclusive ou composite de plusieurs compartiments tissulaires de la prostate que sont l’épithélium, le stroma et les fibres musculaires qui définissent respectivement les composantes glandulaire, fibreuse et musculaire de cette pathologie. Il a récemment été montré que les souris dépourvues en récepteurs nucléaires LXR (Liver‐X‐receptor) α (souris lxrα‐/‐) développent une hypertrophie de la prostate dont les signes histologiques évoquent une HBP de type fibreuse. Par ailleurs, un des traitements de l’HBP, vise à éteindre la signalisation androgénique en inhibant la conversion de la testostérone en son métabolite actif, la dihydrotestostérone (DHT). Le phénotype d’hypertrophie de la prostate pourrait donc également s’expliquer par une altération de la signalisation androgénique dans les souris lxrα‐/‐. Dans ce contexte, notre projet de recherche a été centré sur l’étude du rôle des LXR dans l’apparition de l’HBP dans sa composante glandulaire et l’analyse des relations moléculaires associant les signalisations dépendantes de LXRα et du récepteur des androgènes (AR) au sein de la prostate. Le phénotype d’HBP observé dans les souris lxrα‐/‐ résulte d’altérations importantes de l’homéostasie de l’épithélium qui miment la composante glandulaire : 1) une activité sécrétoire accrue ; 2) une altération des processus de sécrétion associée à une altération de l’expression des gènes codant des protéines du transport vésiculaire ; 3) une réponse altérée de certains gènes androgéno‐dépendants associée à une hypersensibilité aux androgènes ; 4) des modifications du réseau paracrine reliant le stroma et l’épithélium. Au final, ces travaux définissent LXRα comme un acteur clé de l’homéostasie prostatique et ouvrent des pistes intéressantes pour la compréhension de l’étiologie de l’HBP chez l’homme. Ces résultats montrent qu’il est possible de moduler la réponse androgénique de la prostate en ciblant LXRα. Ainsi, à plus long terme, l’activation pharmacologique de LXRα constitue une piste potentielle dans le traitement de l’HBP. / Benign prostate hyperplasia (BPH) is a very common prostatic disorder that affects 50% of men after 60 years. In BPH, prostate enlargement causes urinary disorders. BPH is characterized by a hypertrophy of several tissue compartments such as the epithelium, stroma and/or muscle fibers. Hence, three main forms of BPH have been described : glandular, fibrous and muscular form. It has been recently shown that LXR (Liver‐X‐receptor) α (lxrα‐/‐) mice develop a prostate enlargement with histological signs of fibrous BPH. Inhibition of testosterone conversion into DHT is one the most effective pharmacological treatment of BPH. Thus, the lxrα‐/‐ prostate phenotype could be in part due to an alteration of androgen signaling. In this context, the aim of this work was to study the role of LXR in glandular BPH development and to understand the relationships between LXRα and the androgen receptor (AR) dependent signaling pathway in prostate. The prostate enlargement observed in lxrα‐/‐ mice results from major alterations in epithelium homeostasis mimicking the glandular alteration of BPH : 1) increase of secretory activity ; 2) alteration of the secretory process associated with altered expression of vesicular transport protein encoding genes ; 3) a disruption in the response of androgen‐dependent genes associated with androgen hypersensitivity ; 4) changes in the paracrine network between stroma and epithelium. Finally, this work defines LXRα as a key player in prostate homeostasis and opens interesting way to the understanding of BPH etiology. These results show that targeting LXRα modulate the prostate androgenic response. Thus, pharmacological activation of LXRα could constitute a new option for the treatment of BPH.

A finite difference based solution method is derived for the velocity-pressure formulation of the two-dimensional incompressible Navier-Stokes equations. The method is proven stable using the energy method, facilitated by SBP operators, for characteristic and Dirichlet boundary condition implemented using the SAT technique. The numerical experiments show the utility of high-order finite difference methods as well as emphasize the problem of pressure boundary conditions. Furthermore, we demonstrate that a discretely divergence free solution can be obtained by use of the projection method.

In this thesis a high order finite difference scheme is derived and implemented solving the shallow water equations using the SBP-SAT method. This method was tested against various benchmark problems were convergence was verified. The shallow water equations were also solved on a multi-block setup representing a tsunami approaching a shoreline from the ocean. Experiments show that a bottom topography with many spikes provides a dispersing effect on the incoming tsunami wave. Higher order convergence is not guaranteed for the multi-block simulations and could be investigated further in a future study.

The study examines the Projection method and the simultaneousapproximation-term (SAT) method as boundary treatment for the dynamic beam equation using summation-by-parts (SBP) operators for handling the inner domain. The methods are examined for both the homogeneous constant coefficient case, and the inhomogeneous piecewise constant coefficient case with a coupled interface. The outer boundaries are handled by SAT or Projection, the coupled interfaced is handled by Projection or a mix between Projection and SAT. Solutions are integrated in time with finite central difference schemes and compared to analytical solutions. A convergence study is conducted with respect to the spatial discretization to measure the accuracy, and the stability is examined by numerical simulations of the CFL-condition. The study shows that Projection has the same accuracy as SAT for most boundary conditions while allowing for a larger timestep. A discontinuity in the medium is found to be handled equally accurate by Projection and the Projection and SAT mixture for all but one case studied, where the mixture was slightly more accurate.

No description available.

I [Mattson, Journal of Scientific Computing 51.3 (2012), s. 650–682] konstruerades partialsummeringsoperatorer för finita differensapproximationer av andraderivator med variabla koefficienter. Vi tillämpar framgångsrikt dessa operatorer på vågekvationen i två dimensioner med diskontinuerliga koefficienter, utan särskild behandling av diskontinuiteten. Närmare bestämt undersöks (i) operatorernas fel och konvergensordning relativt ”korrekt” hantering av diskontinuiteter genom blockuppdelning med kopplingstermer; (ii) ifall mycket komplicerade koefficienter orsakar instabilitet eller icke-fysikaliska fel. Vi visar att hoppet i våghastighet i simuleringen sker ett antal punkter ifrån hoppet i koefficienter, där antalet punkter beror på operatorernas ordning och storleken av hoppet i koefficienter. I (i) får dessa två faktorer plus blockets form och antalet punkter en stor påverkan på både storleken av felet, samt metodens konvergensordning som varierar från ca 1–2,5. Annars sker i både (i) och (ii) inget större icke-fysikaliskt fel eller instabilitet, vilket gör denna relativt enkla metod tillämpningsbar på komplexa verklighetsbaserade problem.

This project studied two different ways of imposing boundary conditions weakly with the finite difference summation-by-parts (SBP) operators. These operators were combined with the boundary handling methods of simultaneous-approximation-terms (SAT) and the Projection to impose homogeneous Neumann and Dirichlet boundary conditions. The convergence rate of both methods was analyzed for different boundary conditions for the one-dimensional (1D) Schrödinger equation, without potential, which resulted in both methods performing similarly. A multi-block discretization was then implemented and different combinations of SBP-SAT and SBP-Projection were applied to impose inner boundary conditions of continuity between the blocks. A convergence study of the different methods of imposing the inner BC:s was conducted for the 1D Schrödinger equation without potential. The resulting convergence was the same for all methods and it was concluded that they performed similarly. Methods involving SBP-Projection had the slight advantage of faster computation time. Finally, the 1D Gross-Pitaevskii equation (GPE) and the 1D Schrödinger equation were analyzed with a step potential. The waves propagating towards the potential barrier were in both cases partially transmitted and partially reflected. The waves simulated with the Schrödinger equation dispersed, while the solitons simulated with the GPE kept their shape due to the equations reinforcing non-linear term. The bright soliton was partly transmitted and partly reflected. The dark soliton was either totally reflected or totally transmitted.

Background: Hypertension is defined as an elevated systolic blood pressure (SBP ≥ 140 mmHg), or an elevated diastolic blood pressure (DBP ≥ 90 mmHg). The prevalence of hypertension is high in obese population. The potential effects of inadequate calcium intake on hypertension are receiving growing attention. The aim of the study was to examine the association between calcium intake and hypertension among obese adults. Methods: A total of 14,856 obese adults aged 20 years or older were obtained from the 1999-2010 National Health and Nutrition Examination Survey. Analysis of variance was used to examine if there was a relationship between calcium intake and blood pressure, SBP or DBP. Multiple logistic regressions were used to examine the association between calcium intake and hypertension after adjusting for potential confounders (energy intake, age, race, education level, alcohol use, smoking, and diabetes). Results: Prevalence of hypertension decreased with an increasing quartile of calcium intake (p < 0.0001). Multiple logistic regression showed that lowest quartile of calcium intake was associated with an increased risk of elevated SBP and elevated DBP (Odds Ratio (OR) =1.332, 95% Confidence Interval (CI): 1.084-1.636; OR=1.700, 95% CI: 1.234-2.342, respectively). Compared with adults in the highest quartile of calcium intake, those in lowest quartile had 1.4 times increased risk of hypertension (OR=1.400, 95% CI: 1.157-1.694). Conclusion: Our study provides support of research perspective that inadequate calcium intake may increase the risk of hypertension, high SBP, or high BDP among obese adults. Further studies are needed to understand physiological mechanism. Increasing the calcium intake in obese adults can be considered as a strategy to prevent hypertension.