HEAT EQUATION: CONVERGENCE OF DISCRETE VERSION TO CONTINUOUS VERSION - Anindita, Wilsara
Heat ow in homogeneous continuum medium can be formulated through an initial condition problem and boundary condition from heat equation. The solution of this problem can be expressed with deterministic and probabilistic methods. Because the heat equation has a unique solution, then both of these approaches only give dierent representation of the same solution. One of simple numerical methods to solve heat equation is discretization of nite dierence schemes. In this approach, we obtain initial and boundary value problem, discrete heat equation and discrete laplacian operator. Discrete problem can be solved with deterministic and probabilistic methods. The probabilistic method involves concept of random walk. By rening the size of the grid, it is expected that this discrete problem will converge to the real problem. In this case, we will discuss about denition and conditions that discrete boundary value problem converge to continuous boundary problem. The main of this problem is how random walk converge to brownian motion with scaling limit process. Furthermore, we will be derive that the condition of discrete problem converge to continuous problem.