Explicit Multistep Mixed Finite Element Method for RLW Equation

and Applied Analysis 3 Table 1: Solitary wave Amp. 0.3 and the errors in L2 and L∞ norms for u, Q 1 , Q 2 , and Q 3 at t = 20, h = 0.125, Δt = 0.1, and −40 ≤ x ≤ 60. Method Time Q 1 Q 2 Q 3 L 2 for u L∞ for u Our method 0 3.9797 0.8104 2.5787 0 0 4 3.9797 0.8104 2.5786 3.6304e − 004 5.2892e − 005 8 3.9797 0.8104 2.5786 7.2873e − 004 5.8664e − 005 12 3.9797 0.8104 2.5787 1.0817e − 003 6.3283e − 005 16 3.9795 0.8104 2.5787 1.4186e − 003 6.8001e − 005 20 3.9790 0.8103 2.5785 1.7396e − 003 7.7154e − 005 [20] 20 3.9800 0.8104 2.5792 1.7569e − 003 6.8432e − 004 [21] 20 3.9820 0.8087 2.5730 4.688e − 003 1.755e − 003 [22] 20 3.9905 0.8235 2.6740 2.157e − 003 — [23] 20 3.9616 0.8042 2.5583 0.018e − 003 1.566e − 003 [24] 20 3.9821 0.8112 2.5813 0.511e − 003 0.198e − 003 Table 2: Convergence order and error in L2 norm for u of time with h = 0.125 and c = 0.1. Time Δt = 0.4 Δt = 0.2 Δt = 0.1 Order (0.2/0.4) Order (0.1/0.2) 4 5.3805e − 003 1.4267e − 003 3.6304e − 004 1.9151 1.9745 8 1.1688e − 002 2.9411e − 003 7.2873e − 004 1.9906 2.0129 12 1.7830e − 002 4.3997e − 003 1.0817e − 003 2.0188 2.0241 16 2.3751e − 002 5.7916e − 003 1.4186e − 003 2.0360 2.0295 20 2.9434e − 002 7.1148e − 003 1.7396e − 003 2.0486 2.0321 We take linear basis functions defined as follows: L 1 = 1 − μ, L 2 = μ, (11) and then the variables u and q over the element [x k , x k+1 ] are written as