Numerical Algorithms For High Order Derivative Equations And Their Applications

In this thesis, the main work includes using local discontinuous Galerkin (LDG) method to solve partial differential equations and utilizing a partial differential equation-based regularization algorithm to study "de-noising" problems in biomedical-graphy. This thesis can be divided into five parts.In the first part, we mainly study how to simulate the quantum transport phe-nomenon on the quantum directional coupler with complicated geometry structure. This phenomenon can be interpreted by a steady state Schrodinger equation. We use the minimal dissipation LDG method to solve this equation. In order to make the scheme stable, we add artificial penalty in the flux only at the boundary. Moreover, according to the property of quantum transport phenomenon, the frequency change mainly reflect in the y direction, we use the LDG method not only based on polynomial basis func-tions but also based on exponential basis functions to reduce the computational cost. Numerical experiments demonstrate the capability of numerical schemes.In the second part, we present the energy conserving LDG method for the non-linear Schrodinger equation with wave operator (NLSW). Energy conservation is an important property of the NLSW and the nonconservative scheme may easily lead nu-merical solutions blow up, consequently we use the conservative numerical scheme with LDG spatial discretization and Crank-Nicholson temporal discretization, and we prove the fully scheme is conservative. In addition, we also give the proof of opti-mal error estimates of LDG method for linear case. Numerical results demonstrate the advantage of the conservative scheme in long time simulation.In the third part, we adapt the positivity-preserving high order LDG method to solve the parabolic equation with blow-up solutions. If the initial condition and the source term are positive, the exact solution should be positive by a maximum principle. High order scheme without positivity-preserving limiter may obtain wrong blow-up time and blow-up region. In addition, due to the Dirichlet boundary conditions, we add a penalty term in the flux at the boundary to obtain the optimal convergent accuracy. Numerical results show that positivity preserving LDG method can capture the blow-up time and blow-up region precisely.In the fourth part, we study a partial differential equation-based regularization de- noising algorithm which is used to ultrasound breast elastography. The conventional clinical equipment can only obtain relatively accurate axial (parallel to the acoustic beam) displacement estimation while the quality of the lateral (perpendicular to the beam) displacement estimation is lower. However, obtaining accurate ultrasonically-estimated displacements along both axial and lateral directions is an important task for various clinical elastography applications, such as modulus reconstruction and temper-ature imaging. Hence, we mainly discuss how to improve lateral speckle tracking accu-rate through using conventional ultrasound echo data acquired by clinical equipment. This algorithm has been tested using computer-simulated data, a tissue-mimicking phantom and in vivo data.In the fifth part, we study how to reduce the noise in magnetic resonance angiog-raphy in biomedical fields. For phase-contrast magnetic resonance angiography(PC-MRA), time-resolved3D PC-MRA has only been used to determine basic information such as aneurysm dimensions and flow rates. However, measurement errors and poten-tial artifacts can adversely affect accuracy of PC-MRI/PC results. But characterizing and interpreting complex flow that occurs in human intracranial aneurysms needs more accurate angiography. Hence, we use the partial differential equation-based regulariza-tion algorithm to reduce the noise. Computer-simulated data shows the capability of this algorithm.