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1	Stabila approximationer av ett illaställt Cauchy-problem för värmeledningsekvationen i flera lager / Stable Approximations for a Cauchy Problem for the Heat Equations Gustafsson, Fredrik January 2014 (has links) I den här kandidatuppsatsen betraktar vi ett illa ställt Cauhy-problem för värmeekvationen i tunna väggar. Problemet kan matematiskt formuleras som: givet brusiga mätningar u x(L, t) och u(L, t) av ux(L, t) och u(L, t) längs linjen x = L bestäm lösningen för u(x, t) för 0 ≤ x, där u uppfyller värmeledningsekvationen ut = ((x)ux)x. Problemet dyker upp i många tillämpningar när man vill uppskatta en temperatur men att en direkt mätning inte låter sig göras. Man mäter en temperatur på ett annat ställe och beräknar temperaturen i den punkt man är intresserad av. Tyngdpunkten i uppsatsen ligger i att undersöka det inversa värmeproblemet i en vägg med flera lager. Det modelleras med att (x) antas vara styckvis konstant. Problemet undersöks med Fourieranalys. Analysen visar att problemet är extremt illaställt och att det beror på förstärkning av höga frekvenser från mätbruset. Vi presenterar numeriska metoder för att regularisera och lösa problmet. Numersiska exempel som illustrar illaställdheten och regulariserade lösningar ges. Inversa problem illaställt regularisering finita differenser värmeöverföring
2	Direct and inverse methods for waveguides and scattering problems in the time domain / Abenius, Erik, January 2005 (has links) Diss. (sammanfattning) Uppsala : Uppsala universitet, 2005. / Härtill 5 uppsatser.
3	Effektiva lösningsmetoder för Schrödingerekvationen : En jämförelse Christoffer, Zakrisson January 2013 (has links) In this paper the rate of convergence, speed of execution and symplectic properties of the time-integrators Leap-Frog (LF2), fourth order Runge-Kutta(RK4) and Crank-Nicholson (CN2) have been studied. This was done by solving the one-dimensional model for a particle in a box (Dirichlet-conditions). The results show that RK4 is the fastest in achieving higher tolerances, while CN2 is the fastest in achieving lower tolerances. Fourth order corrections of LF (LF4)and CN (CN4) were also studied, though these showed no improvements overLF2 and CN2. All methods were shown to exhibit symplectic behavior. Finita differenser SBP-SAT Schrödingerekvationen Leap-Frog Crank-Nicholson Runge-Kutta Computer and Information Sciences Data- och informationsvetenskap Computational Mathematics Beräkningsmatematik
4	Numerical simulation of acoustic wave propagation with a focus on modeling sediment layers and large domains Estensen, Elias January 2022 (has links) In this report, we study how finite differences can be used to simulate acoustic wave propagation originating from a point source in the ocean using the Helmholtz equation. How to model sediment layers and the vast size of the ocean is studied in particular. The finite differences are implemented with summation by parts operators with boundary conditions enforced with simultaneous approximation terms and projection. The numerical solver is combined with the WaveHoltz method to improve the performance. Sediment layers are handled with interface conditions and the domain is artificially expanded using absorbing layers. The absorbing layer is implemented with an alternative approach to the super-grid method where the domain expansion is accomplished by altering the wave speed rather than with coordinate transformations. To isolate these issues, other parameters such as variations in the ocean floor are neglected. With this simplification, cylindrical coordinates are used and the angular variation is assumed to be zero. This reduces the problem to a quasi-three-dimensional system. We study how the parameters of the alternative absorbing layer approach affect its quality. The numerical solver is verified on several test cases and appears to work according to theory. Finally, a semi-realistic simulation is carried out and the solution seems correct in this setting. Helmholtz wave equation acoustics ocean acoustics numerical analysis computational science finite differences summation by parts sbp waveholtz cylindrical coordinates sediment layers interface super-grid super grid absorbing layer Helmholtz vågekvationen akustik undervattensakustik numerisk analys beräkningsvetenskap finita differenser summation by parts sbp cylindriska koordinater waveholtz sedimentskikt interface super-grid absorberande lager Computational Mathematics Beräkningsmatematik

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