Numerical Methods Library for OCTAVE

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X equispaced grid of NX intervals.U solution array.Notice that we use beuler to integrate each ODE of this system. In fact if we are computingthe solution of the heat equation for example. After the finite difference approximationwe obtain the system y ′ = Ay with A = tridiag(1,-2,1). The Jacobian matrix A of this systemhas eigenvalues between 4c/(∆x)2 and 0, which makes the ODE very stiff as the spatial meshsize ∆x becomes small. This stiffness, which is typical of ODEs derived from PDEs in thismanner, must be taken into account in choosing an ODE method for solving the semidiscretesystem.3.10.2 2-D solver for Advection equationAdvection equation :u t + cu x = 0 (20)[T, X, U] = PDE_ADVEC_EXP(N, DX, K, DT, C, F) solves the advection equationD U/DT = -C D U/DX with the explicit method on [0,1]x[0,1]. Initial conditionis U(0,X) = F. C is a positive constant. N is the number of space integrationintervals and K is the number of time-integration intervals. DX is the sizeof a space integration interval and DT is the size of time-integration intervals.ParametersNX number of space integration intervals.DX size of a space integration interval.NT number of time-integration intervals.DT size of time-integration intervals.C positive constant.F initial condition U(0,X) = F(X).ReturnsT grid of NT intervals.X grid of NX intervals.U solution array.[T, X, U] = PDE_ADVEC_IMP(N, DX, K, DT, C, F) solves the advection equationD U/DT = -C D U/DX with the implicit method on [0,1]x[0,1]. Initial conditionis U(0,X) = F. C is a positive constant. N is the number of space integrationintervals and K is the number of time-integration intervals. DX is the sizeof a space integration interval and DT is the size of time-integration intervals.ParametersNX number of space integration intervals.DX size of a space integration interval.NT number of time-integration intervals.DT size of time-integration intervals.C positive constant.27
Figure 3: Solution of the Advection equation with x ∈ [0, 1] and t ∈ [0, 0.6].F initial condition U(0,X) = F(X).ReturnsT grid of NT intervals.X grid of NX intervals.U solution array.3.10.3 2-D solver for Heat equationHeat equation :u t = cu xx (21)[T, X, U] = PDE_HEAT_EXP(N, DX, K, DT, C, F, ALPHA, BETA) solves the heatequation D U/DT = C D2U/DX2 with the explicit method on [0,1]x[0,1]. Initialcondition is U(0,X) = F and boundary conditions are U(t,0) = ALPHA and U(t,1)= beta. C is a positive constant. N is the number of space integrationintervals and K is the number of time-integration intervals. DX is the sizeof a space integration interval and DT is the size of time-integration intervals.ParametersNX number of space integration intervals.DX size of a space integration interval.NT number of time-integration intervals.DT size of time-integration intervals.F initial condition U(0,X) = F(X).C positive constant.28
Page 3 and 4: 3.8.2 Implicit Euler . . . . . . .
Page 6 and 7: • Euler• Implicit Euler• Modi
Page 8 and 9: X0 initial guess.ReturnsX computed
Page 10 and 11: 3.2.1 Normal equation[X, RES] = NOR
Page 12 and 13: X0 initial point.ITMAX maximal numb
Page 14 and 15: 3.4.1 Monomial basisP = ITPOL_MONOM
Page 16 and 17: N number of subdivisions.ReturnsRES
Page 18 and 19: 3.6.4 Orthogonal iteration[LAMBDA,
Page 20 and 21: H f (xk)s k = f (x k ) (12)for s k
Page 22 and 23: 3.8.1 Euler[TT,Y] = ODE_EULER(ODEFU
Page 24 and 25: condition Y0 using the fourth-order
Page 26 and 27: Figure 2: Shooting method for a two
Page 30 and 31: ReturnsT grid of NT intervals.X gri
Page 32: A, BC, D rectangle (A,B)x(C,D) wher

Numerical Methods Library for OCTAVE

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