In this work, the numerical solutions of advection-diffusion equation are investigated through the finite element method. The quartic-trigonometric tension (QTT) B-spline which presents advantages over the well-known existing B-splines is adapted as the base of the numerical algorithm. Space integration of the model partial differential equation is achieved through QTT B-spline Galerkin method. The resultant system of time-dependent differential equations is integrated using the Crank-Nicolson technique. The stability of the current scheme is accomplished and proved to be unconditionally stable. Simulation of several sample problems are carried out for verification of the proposed numerical scheme. Solutions obtained by numerically computed scheme are compared to the existing literature.

在这项工作中，通过有限元方法研究了对流扩散方程的数值解。表现出优于已知的现有B样条的优势的四次方三角张力（QTT）B样条被用作数值算法的基础。模型偏微分方程的空间积分是通过QTT B样条Galerkin方法实现的。使用Crank-Nicolson技术对所得的时变微分方程组进行积分。当前方案的稳定性得以实现，并证明是无条件稳定的。为了验证所提出的数值方案，对几个样本问题进行了仿真。将通过数值计算方案获得的解与现有文献进行比较。