7 added 14 characters in body; edited tags edited Apr 21 '17 at 7:59 xzczd 29.8k66 gold badges8484 silver badges276276 bronze badges (*initialisation of parameters and initial conditions*) Sigma0 = 0.002; g0 = 0.001; L = 1; \[Rho]ρ = 1; Ei = 1; c =Sqrt[Ei/\[Rho]];ρ]; epsi = 10^-7.; qu[x_] := Piecewise[{{(-1 Sigma0/Ei) x, 0 <= x <= L/2}, {(-1 Sigma0/Ei) L/2, L/2 <= x <= L}}]; qv[x_] := Piecewise[{{0, 0 <= x <= L/2}, {(c Sigma0/Ei), L/2 <= x <= L}}]; (*Building of a periodic Solution according to the wave equation in Dirichlet-Neumann Boundary condition*) u0[x_] := qu[x]; v0[x_] := qv[x]; uif = NDSolveValue[{D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] == 0, u[x, 0] == u0[x], (D[u[x, t], t] /. t -> 0) == v0[x], (D[u[x, t], x] /. x -> L) == 0, u[0, t] == 0}, u, {x, 0, L}, {t, -10, 10}]; tc = t /. FindRoot[uif[L, t] - g0, {t, 0.1}]; (*Plot of the solution at a time tc in purple and the initial condition*) Show[Plot[uif[x, 0], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Red], Plot[uif[x, tc], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Purple], PlotRange -> {{0, L}, All}, AxesLabel -> {"Time", "Displacement"}] NDSolveValue::mxsst: Using maximum number of grid points 10000 allowed by the MaxPoints or MinStepSize options for independent variable x. -NDSolveValue::eerr: Warning: scaled local spatial error estimate of 185.53982144076966at t = 20. in the direction of independent variable x is much greater than the prescribed error tolerance. Grid spacing with 25 points may be too large to achieve the desired accuracy or precision. A singularity may have formed or a smaller grid spacing can be specified using the MaxStepSize or MinPoints method options. NDSolveValue::ndstf: At t == -2.61883, system appears to be stiff. Methods Automatic, BDF, or StiffnessSwitching may be more appropriate. NDSolveValue::mxsst: Using maximum number of grid points 10000 allowed by the MaxPoints or MinStepSize options for independent variable x. NDSolveValue::eerr: Warning: scaled local spatial error estimate of 185.53982144076966 at t = 20. in the direction of independent variable x is much greater than the prescribed error tolerance. Grid spacing with 25 points may be too large to achieve the desired accuracy or precision. A singularity may have formed or a smaller grid spacing can be specified using the MaxStepSize or MinPoints method options. NDSolveValue::ndstf: At t == -2.61883, system appears to be stiff. Methods Automatic, BDF, or StiffnessSwitching may be more appropriate. (*initialisation of parameters and initial conditions*) Sigma0 = 0.002; g0 = 0.001; L = 1; \[Rho] = 1; Ei = 1; c =Sqrt[Ei/\[Rho]]; epsi = 10^-7.; qu[x_] := Piecewise[{{(-1 Sigma0/Ei) x, 0 <= x <= L/2}, {(-1 Sigma0/Ei) L/2, L/2 <= x <= L}}]; qv[x_] := Piecewise[{{0, 0 <= x <= L/2}, {(c Sigma0/Ei), L/2 <= x <= L}}]; (*Building of a periodic Solution according to the wave equation in Dirichlet-Neumann Boundary condition*) u0[x_] := qu[x]; v0[x_] := qv[x]; uif = NDSolveValue[{D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] == 0, u[x, 0] == u0[x], (D[u[x, t], t] /. t -> 0) == v0[x], (D[u[x, t], x] /. x -> L) == 0, u[0, t] == 0}, u, {x, 0, L}, {t, -10, 10}]; tc = t /. FindRoot[uif[L, t] - g0, {t, 0.1}]; (*Plot of the solution at a time tc in purple and the initial condition*) Show[Plot[uif[x, 0], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Red], Plot[uif[x, tc], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Purple], PlotRange -> {{0, L}, All}, AxesLabel -> {"Time", "Displacement"}] NDSolveValue::mxsst: Using maximum number of grid points 10000 allowed by the MaxPoints or MinStepSize options for independent variable x. -NDSolveValue::eerr: Warning: scaled local spatial error estimate of 185.53982144076966at t = 20. in the direction of independent variable x is much greater than the prescribed error tolerance. Grid spacing with 25 points may be too large to achieve the desired accuracy or precision. A singularity may have formed or a smaller grid spacing can be specified using the MaxStepSize or MinPoints method options. NDSolveValue::ndstf: At t == -2.61883, system appears to be stiff. Methods Automatic, BDF, or StiffnessSwitching may be more appropriate. (*initialisation of parameters and initial conditions*) Sigma0 = 0.002; g0 = 0.001; L = 1; ρ = 1; Ei = 1; c =Sqrt[Ei/ρ]; epsi = 10^-7.; qu[x_] := Piecewise[{{(-1 Sigma0/Ei) x, 0 <= x <= L/2}, {(-1 Sigma0/Ei) L/2, L/2 <= x <= L}}]; qv[x_] := Piecewise[{{0, 0 <= x <= L/2}, {(c Sigma0/Ei), L/2 <= x <= L}}]; (*Building of a periodic Solution according to the wave equation in Dirichlet-Neumann Boundary condition*) u0[x_] := qu[x]; v0[x_] := qv[x]; uif = NDSolveValue[{D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] == 0, u[x, 0] == u0[x], (D[u[x, t], t] /. t -> 0) == v0[x], (D[u[x, t], x] /. x -> L) == 0, u[0, t] == 0}, u, {x, 0, L}, {t, -10, 10}]; tc = t /. FindRoot[uif[L, t] - g0, {t, 0.1}]; (*Plot of the solution at a time tc in purple and the initial condition*) Show[Plot[uif[x, 0], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Red], Plot[uif[x, tc], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Purple], PlotRange -> {{0, L}, All}, AxesLabel -> {"Time", "Displacement"}] NDSolveValue::mxsst: Using maximum number of grid points 10000 allowed by the MaxPoints or MinStepSize options for independent variable x. NDSolveValue::eerr: Warning: scaled local spatial error estimate of 185.53982144076966 at t = 20. in the direction of independent variable x is much greater than the prescribed error tolerance. Grid spacing with 25 points may be too large to achieve the desired accuracy or precision. A singularity may have formed or a smaller grid spacing can be specified using the MaxStepSize or MinPoints method options. NDSolveValue::ndstf: At t == -2.61883, system appears to be stiff. Methods Automatic, BDF, or StiffnessSwitching may be more appropriate. 6 added 755 characters in body edited Apr 20 '17 at 21:47 CharlelieB 3133 bronze badges I get those warnings during all the computation : NDSolveValue::mxsst: Using maximum number of grid points 10000 allowed by the MaxPoints or MinStepSize options for independent variable x. -NDSolveValue::eerr: Warning: scaled local spatial error estimate of 185.53982144076966at t = 20. in the direction of independent variable x is much greater than the prescribed error tolerance. Grid spacing with 25 points may be too large to achieve the desired accuracy or precision. A singularity may have formed or a smaller grid spacing can be specified using the MaxStepSize or MinPoints method options. NDSolveValue::ndstf: At t == -2.61883, system appears to be stiff. Methods Automatic, BDF, or StiffnessSwitching may be more appropriate. I get those warnings during all the computation : NDSolveValue::mxsst: Using maximum number of grid points 10000 allowed by the MaxPoints or MinStepSize options for independent variable x. -NDSolveValue::eerr: Warning: scaled local spatial error estimate of 185.53982144076966at t = 20. in the direction of independent variable x is much greater than the prescribed error tolerance. Grid spacing with 25 points may be too large to achieve the desired accuracy or precision. A singularity may have formed or a smaller grid spacing can be specified using the MaxStepSize or MinPoints method options. NDSolveValue::ndstf: At t == -2.61883, system appears to be stiff. Methods Automatic, BDF, or StiffnessSwitching may be more appropriate. 5 deleted 158 characters in body edited Apr 20 '17 at 21:43 anderstood 8,17911 gold badge2020 silver badges6262 bronze badges I am trying to compute the solution to the wave equation using NDSolveValue and singular initial displacement (continuous, but not differentiable in a point). I need to propagate as accurately as possible the singularity of the initial condition through time. However the singularity is "smoothened" during the integration, see below. The red curve is my initial condition in displacement. The otherpurple one is the output at a different time of the computation. According to d'Alembert's solutionAs you can see, they should be the exact symmetry of the red curvespurple curve is perfectly smooth, while it shouldn't. Here you can see a piece of the script I use to get these solutions : (*initialisation of parameters and initial conditions*) Sigma0 = 0.002; g0 = 0.001; L = 1; \[Rho] = 1; Ei = 1; c =Sqrt[Ei/\[Rho]]; epsi = 10^-7.; qu[x_] := Piecewise[{{(-1 Sigma0/Ei) x, 0 <= x <= L/2}, {(-1 Sigma0/Ei) L/2, L/2 <= x <= L}}]; qv[x_] := Piecewise[{{0, 0 <= x <= L/2}, {(c Sigma0/Ei), L/2 <= x <= L}}]; (*Building of a periodic Solution according to the wave equation in Dirichlet-Neumann Boundary condition*) u0[x_] := qu[x]; v0[x_] := qv[x]; uif = NDSolveValue[{D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] == 0, u[x, 0] == u0[x], (D[u[x, t], t] /. t -> 0) == v0[x], (D[u[x, t], x] /. x -> L) == 0, u[0, t] == 0}, u, {x, 0, L}, {t, -10, 10}]; tc = t /. FindRoot[uif[L, t] - g0, {t, 0.1}]; (*Plot of the solution at a time tc in purple and the initial condition*) Show[Plot[uif[x, 0], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Red], Plot[uif[x, tc], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Purple], PlotRange -> {{0, L}, All}, AxesLabel -> {"Time", "Displacement"}] I am trying to compute the solution to the wave equation using NDSolveValue and singular initial displacement (continuous, but not differentiable in a point). I need to propagate as accurately as possible the singularity of the initial condition through time. However the singularity is "smoothened" during the integration, see below. The red curve is my initial condition in displacement. The other one is the output at a different time of the computation. According to d'Alembert's solution, they should be the exact symmetry of the red curves. Here you can see a piece of the script I use to get these solutions : (*initialisation of parameters and initial conditions*) Sigma0 = 0.002; g0 = 0.001; L = 1; \[Rho] = 1; Ei = 1; c =Sqrt[Ei/\[Rho]]; epsi = 10^-7.; qu[x_] := Piecewise[{{(-1 Sigma0/Ei) x, 0 <= x <= L/2}, {(-1 Sigma0/Ei) L/2, L/2 <= x <= L}}]; qv[x_] := Piecewise[{{0, 0 <= x <= L/2}, {(c Sigma0/Ei), L/2 <= x <= L}}]; (*Building of a periodic Solution according to the wave equation in Dirichlet-Neumann Boundary condition*) u0[x_] := qu[x]; v0[x_] := qv[x]; uif = NDSolveValue[{D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] == 0, u[x, 0] == u0[x], (D[u[x, t], t] /. t -> 0) == v0[x], (D[u[x, t], x] /. x -> L) == 0, u[0, t] == 0}, u, {x, 0, L}, {t, -10, 10}]; tc = t /. FindRoot[uif[L, t] - g0, {t, 0.1}]; (*Plot of the solution at a time tc in purple and the initial condition*) Show[Plot[uif[x, 0], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Red], Plot[uif[x, tc], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Purple], PlotRange -> {{0, L}, All}, AxesLabel -> {"Time", "Displacement"}] I am trying to compute the solution to the wave equation using NDSolveValue and singular initial displacement (continuous, but not differentiable in a point). I need to propagate as accurately as possible the singularity of the initial condition through time. However the singularity is "smoothened" during the integration, see below. The red curve is my initial condition in displacement. The purple one is the output at a different time of the computation. As you can see, the purple curve is perfectly smooth, while it shouldn't. Here you can see a piece of the script I use to get these solutions : (*initialisation of parameters and initial conditions*) Sigma0 = 0.002; g0 = 0.001; L = 1; \[Rho] = 1; Ei = 1; c =Sqrt[Ei/\[Rho]]; epsi = 10^-7.; qu[x_] := Piecewise[{{(-1 Sigma0/Ei) x, 0 <= x <= L/2}, {(-1 Sigma0/Ei) L/2, L/2 <= x <= L}}]; qv[x_] := Piecewise[{{0, 0 <= x <= L/2}, {(c Sigma0/Ei), L/2 <= x <= L}}]; (*Building of a periodic Solution according to the wave equation in Dirichlet-Neumann Boundary condition*) u0[x_] := qu[x]; v0[x_] := qv[x]; uif = NDSolveValue[{D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] == 0, u[x, 0] == u0[x], (D[u[x, t], t] /. t -> 0) == v0[x], (D[u[x, t], x] /. x -> L) == 0, u[0, t] == 0}, u, {x, 0, L}, {t, -10, 10}]; tc = t /. FindRoot[uif[L, t] - g0, {t, 0.1}]; (*Plot of the solution at a time tc in purple and the initial condition*) Show[Plot[uif[x, 0], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Red], Plot[uif[x, tc], {x, 0, L}, PlotPoints -> 200, PlotStyle -> Purple], PlotRange -> {{0, L}, All}, AxesLabel -> {"Time", "Displacement"}] 4 added 436 characters in body edited Apr 20 '17 at 21:34 CharlelieB 3133 bronze badges 3 simplified explanation of the question edited Apr 20 '17 at 21:22 anderstood 8,17911 gold badge2020 silver badges6262 bronze badges 2 deleted 264 characters in body edited Apr 20 '17 at 21:11 CharlelieB 3133 bronze badges 1 asked Apr 20 '17 at 20:31 CharlelieB 3133 bronze badges