Hivel2D users manual
Qm + 1 - Qm
m+1
Qm - Qm-1
Qj
1 - α j
1 + α j
j
j
+
≈
(12)
2 t m+1 - t m 2 t m - t m-1
t
where j is the nodal location and m is the time-step. An α equal to 1 results in a first order
backward difference approximation, and an α equal to 2 results in a second-order
backward difference approximation of the temporal derivative.
Solution of the Nonlinear Equations
The system of nonlinear equations is solved using the Newton-Raphson iterative
method. Let Ri be a vector of the nonlinear equations computed using a particular test
function ψ i and using an assumed value of Qj. Ri is the residual error for a particular test
function i. Subsequently, Ri is forced toward zero as:
Rik
Qkj = - Rik
k ∆
(13)
Qj
where the derivatives composing the Jacobian are determined analytically and k is the
iteration number. This system of equations is solved for ∆ Qjk and then an improved
estimate for Qjk+1 is obtained from:
Qkj+1 = Qkj + ∆ Qkj
(14)
This procedure is continued until convergence to an acceptable residual error is obtained.
Model Features
Particular features of HIVEL2D Version 2.0 are as follows:
represent p, q, and h.
7
Chapter 2 HIVEL 2D overview
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