By Jan-Philipp Weiß
Lattice Boltzmann tools are a promising process for the numerical resolution of fluid-dynamic difficulties. We ponder the one-dimensional Goldstein-Taylor version with the purpose to reply to the various questions about the numerical research of lattice Boltzmann schemes. Discretizations for the answer of the warmth equation are offered for a variety of boundary stipulations. balance and convergence of the ideas are proved through making use of strength estimates and specific Fourier representations.
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Extra resources for Numerical Analysis of Lattice Boltzmann Methods for the Heat Equation on a Bounded Interval
2) to determine Neu Neu (·) + rL (·) ∂x r (·, xL ) := g (·, xL ) + ν 2 ∂t rL ∂x r (·, xR ) := g (·, xR ) + ν 2 Neu ∂t rR (·) + Neu rR (·) in (0, T ), in (0, T ). 3). For the inﬂow problem there is no information on r or j at the boundaries. For u and v only the inﬂow values (and their derivatives) are known. Hence, we cannot determine a solution of this problem by using this approach. 2. The Telegraph Equation We have to consider solutions of the telegraph equation L s := ν 2 ∂t2 s + ∂t s − ν∂x2 s = h in ΩT .
1]. 7. 29). Proof. See Ref. 1]. 7 can be written as Fourier series rNeu (t, x) = 1 2 (r0Neu , φNeu 0 )+ ∞ + rRob (t, x) = k=1 ∞ k=1 t 0 Neu Neu (fLR (s, ·), φNeu 0 ) ds φ0 (x) −λk t + (r0Neu , φNeu k )e t 0 t 2 −νμk t (r0Rob , φRob + k )e 0 Neu −λk (t−s) (fLR (s, ·), φNeu ds φNeu k )e k (x), 2 Rob (fLR (s, ·), φRob )e−νμk (t−s) ds φRob k k (x). From the Fourier representations of the solutions for the Neumann problem we ﬁnd 22 2. 8. 6, we have rNeu (t, ·) 2 0 ≤ 2 r0Neu 2 0 + max 2t, ν|rNeu (t, ·)|21 ≤ 2ν|r0Neu |21 e−2λ1 t + t 1 λ1 t 0 Neu fLR (s, ·) Neu fLR (s, ·) 0 2 0 2 0 ds, ds.
13). A Hilbert basis in HP1 (Ω), consisting of orthogonal eigenP functions, is obtained by φP k k≥0 ∪ ψk k≥1 with φP k (x) := 2 cos |Ω| 2kπ (x − xL ) |Ω| for k = 0, 1, . . , ψkP (x) := 2 sin |Ω| 2kπ (x − xL ) |Ω| for k = 1, 2, . . 17). 12. 13). Proof. Since we have H01 (Ω) ⊆ HP1 (Ω) ⊆ H 1 (Ω) we can use Ref. 1]. 24 2. THE HEAT EQUATION The Fourier series of the periodic solution reads 1 r (t, x) = 2 P (r0 , φP 0) ∞ t + 0 P (f (s, ·), φP 0 ) ds φ0 (x) t −4λk t + (r0 , φP k )e + k=1 ∞ 0 t (r0 , ψkP )e−4λk t + + −4λk (t−s) (f (s, ·), φP ds φP k )e k (x) 0 k=1 (f (s, ·), ψkP )e−4λk (t−s) ds ψkP (x).