Legendre Semi-Discrete Pseudo-Spectral Method For Hyperbolic Equations And Its Parallel Methods

In this thesis, the main idea is to solve the following one order hyperbolic equation, which has the divergence form, with semi-discrete Legendre method.Introduce operatorIt is quite easy to see that v_x = -u. Let u to be the solution of (2.1), then v = Gu satisfiesCanuto and Quarteroni gave us the Lengendre pseudo-spectral approximation of the linear problem thatwith Gauss-Lobatto node. Let u_c to be Legendre approximation. With the general b(x), they achieved the estimation thatHere, we use Gauss-Radau node to prove that the estimation could be improved, which is one of the main aims. Define the discrete inner-product:and operators:It is easy to prove that P_c(x_j) = u(x_j), 0≤j≤N.We use the following form to be the approximation of (2.1), To find u_c∈C¹((0,T);S_N (I)), which satisfiesWhile is the coefficient of the integral.Besides, we also use the following lemma:Lemma 1^([2])For any real numbersσandμ, there exists constant C, so that the following inequalities come into existence,Lemma 2^([1],[5]) There exists operatorΠ_N : H¹(I)→S_N(I), so thatandLemma 3(Fubini Theorem) Suppose f(x, y) is R^p+q = R^P×R^q the function could be integrable.Then (1)for almost all x∈R^p, f(x, y) is the function of y;(2)for the function which has definition almost everywhereis integrable in R^p;(3)there exists equalityLemma 4 Suppose v∈H_E¹(I)∩H^σ(I), thenLemma 5 Suppose v∈S_N, thenLemma 6 Suppose e∈H_E¹;(I), thenLemma 7 Suppose u,Ï…S_n+1, thenLemma 8(Gronwall Inequalities) Supposeφ(t) and g(t)为[0,T] to be the non-negative continuous functions, andφ(t) is differentiable in [0,T]. If there exists constantα≥0, so that for any t∈(0,T), andor equivalently,Then With the above lemma, we could prove the error estimation theorem:Theorem 4.1 Separately suppose u and u_c to be the solution of (2.1) and (2.5). Function b(x)∈C^σ(I), and satisfies the condition (H). AndThen we have the following estimation:WhileFunction u_c could be expressed asWhile are the base functions of Lagrange interpolation functions. To the equation(2.1), there are two forms.Firstly, The other one,By these methods, in order to get u_c, we only need to solve N + 1 ordinary differential equations with variable t and the unknown functions (x_i,t).Parallel Method : Firstly, to divide the original domainΩ= [—1,1] x [0,T] to sub-domains. In the space direction x, we make h = 2/K, and in the time direction t, we makeâ–³t =T/M, while K, M are both positive numbers. Then Space domain I = [-1,1] could be divided to the form thatAnd the time domain A = [0, T] could be divided asLetBy the parallel method, in order to find the solution u_c^k,m in the domainΩ_k,m, we only have to know the solutions u_c^(k-1),m , inΩ_(k-1),m, and u_c ^(k,(m-1)) , inΩ_k,(m-1). Their relationship could be described as the following picture. The computation sequence could be seen in the following picture. In the sub-domain,Ω_k,m, the solution u_c^k,m satisfies the following equation,While, is the coefficient of the integral, andBy the transformation of coordinates(5.7) could be changed to be the equations, which have x as their variable. So the problem in I_k = [yk-1),yk] could be changed to the problem in domain I = [—1,1]. And we could solve the the problem as (5.3) or (5.5).At last, there is an example based on this method. From the data of the the computation, we could see that the approximation is nearly exact to the real solution.