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I have a model of a system which consists of diffusing reactants and intermediates. For each variable, $u_i$, the final representative equation looks like the standard reaction-diffusion form:

$$\frac{\partial}{\partial t}u_i(x,t)=f_i(\bar{u}(x,t))-D\frac{\partial^2}{\partial x^2}u_i(x,t)$$

where $f_i$ is a non-linear function that denotes the reaction term. I am using the finite difference method (FDM) to discretize the spatial derivative with Neumann boundary conditions: $$\frac{\partial u_i}{\partial x}{\bigg|}_{x=0, x=L} = 0$$

using the central difference formula:

$$u_i''(x)\approx\frac{u_i(x-h) - 2u_i(x) + u_i(x+h)}{h^2}.$$ After reducing the PDE to an ODE, I am using MATLAB's ode45 (RK4 based solver) to numerically integrate the equations in time.

I have not worked on PDEs much and online resources on FDM (such as even wikipedia), depict simultaneous discretization of the time derivative, also.

As far as I understand, the discretization shown in these cases, depict the simple Euler integration (either explicit or implicit). Even the Crank-Nicolson scheme is shown like that.

Is my approach correct (discretize spatial derivative and solve using an optimized ode solver)? Can there be numerical stability issues?

I have a model of a system which consists of diffusing reactants and intermediates. For each variable, $u_i$, the final representative equation looks like the standard reaction-diffusion form:

$$\frac{\partial}{\partial t}u_i(x,t)=f_i(\bar{u}(x,t))-D\frac{\partial^2}{\partial x^2}u_i(x,t)$$

where $f_i$ is a non-linear function that denotes the reaction term. I am using the finite difference method (FDM) to discretize the spatial derivative with Neumann boundary conditions: $$\frac{\partial u_i}{\partial x}{\bigg|}_{x=0, x=L} = 0$$

using the central difference formula:

$$u_i''(x)\approx\frac{u_i(x-h) - 2u_i(x) + u_i(x+h)}{h^2}.$$ After reducing the PDE to an ODE, I am using MATLAB's ode15s (RK4 based solver) to numerically integrate the equations in time.

I have not worked on PDEs much and online resources on FDM (such as even wikipedia), depict simultaneous discretization of the time derivative, also.

As far as I understand, the discretization shown in these cases, depict the simple Euler integration (either explicit or implicit). Even the Crank-Nicolson scheme is shown like that.

Is my approach correct (discretize spatial derivative and solve using an optimized ode solver)? Can there be numerical stability issues?

I have a model of a system which consists of diffusing reactants and intermediates. For each variable, $u_i$, the final representative equation looks like the standard reaction-diffusion form:

$$\frac{\partial}{\partial t}u_i(x,t)=f_i(\bar{u}(x,t))-D\frac{\partial}{\partial x^2}u_i(x,t)$$

Where $f_i$ is a non-linear function that denotes the reaction term. I am using the finite difference method (FDM) to discretize the spatial derivative with Neumann boundary conditions: $$\frac{du_i}{dx}{\bigg|}_{x=0/L} = 0$$

using the central difference formula:

$$u_i''(x)\approx\frac{u_i(x-h) + 2u_i(x) - u_i(x+h)}{h^2}$$ After reducing the PDE to an ODE, I am using MATLAB's ode45 (RK4 based solver) to numerically integrate the equations in time.

I have not worked on PDEs much and online resources on FDM (such as even wikipedia), depict simultaneous discretization of the time derivative, also.

As far as I understand, the discretization shown in these cases, depict the simple Euler integration (either explicit or implicit). Even the Crank-Nicolson scheme is shown like that.

Is my approach correct (discretize spatial derivative and solve using an optimized ode solver)? Can there be numerical stability issues?

I have a model of a system which consists of diffusing reactants and intermediates. For each variable, $u_i$, the final representative equation looks like the standard reaction-diffusion form:

$$\frac{\partial}{\partial t}u_i(x,t)=f_i(\bar{u}(x,t))-D\frac{\partial^2}{\partial x^2}u_i(x,t)$$

where $f_i$ is a non-linear function that denotes the reaction term. I am using the finite difference method (FDM) to discretize the spatial derivative with Neumann boundary conditions: $$\frac{\partial u_i}{\partial x}{\bigg|}_{x=0, x=L} = 0$$

using the central difference formula:

$$u_i''(x)\approx\frac{u_i(x-h) - 2u_i(x) + u_i(x+h)}{h^2}.$$ After reducing the PDE to an ODE, I am using MATLAB's ode45 (RK4 based solver) to numerically integrate the equations in time.

I have not worked on PDEs much and online resources on FDM (such as even wikipedia), depict simultaneous discretization of the time derivative, also.

As far as I understand, the discretization shown in these cases, depict the simple Euler integration (either explicit or implicit). Even the Crank-Nicolson scheme is shown like that.

Is my approach correct (discretize spatial derivative and solve using an optimized ode solver)? Can there be numerical stability issues?

I have a model of a system which consists of diffusing reactants and intermediates. For each variable, $u_i$, the final representative equation looks like the standard reaction-diffusion form:

$$\frac{\partial}{\partial t}u_i(x,t)=f_i(\bar{u}(x,t))-D\frac{\partial}{\partial x^2}u_i(x,t)$$

Where $f_i$ is a non-linear function that denotes the reaction term. I am using the finite difference method (FDM) to discretize the spatial derivative with Neumann boundary conditions: $$\frac{du_i}{dx}{\bigg|}_{x=0/L} = 0$$

After reducing the PDE to an ODE, I am using MATLAB's ode45 (RK4 based solver) to numerically integrate the equations in time.

I have not worked on PDEs much and online resources on FDM (such as even wikipedia), depict simultaneous discretization of the time derivative, also.

As far as I understand, the discretization shown in these cases, depict the simple Euler integration (either explicit or implicit). Even the Crank-Nicolson scheme is shown like that.

Is my approach correct (discretize spatial derivative and solve using an optimized ode solver)? Can there be numerical stability issues?

I have a model of a system which consists of diffusing reactants and intermediates. For each variable, $u_i$, the final representative equation looks like the standard reaction-diffusion form:

$$\frac{\partial}{\partial t}u_i(x,t)=f_i(\bar{u}(x,t))-D\frac{\partial}{\partial x^2}u_i(x,t)$$

Where $f_i$ is a non-linear function that denotes the reaction term. I am using the finite difference method (FDM) to discretize the spatial derivative with Neumann boundary conditions: $$\frac{du_i}{dx}{\bigg|}_{x=0/L} = 0$$

using the central difference formula:

$$u_i''(x)\approx\frac{u_i(x-h) + 2u_i(x) - u_i(x+h)}{h^2}$$ After reducing the PDE to an ODE, I am using MATLAB's ode45 (RK4 based solver) to numerically integrate the equations in time.

I have not worked on PDEs much and online resources on FDM (such as even wikipedia), depict simultaneous discretization of the time derivative, also.

As far as I understand, the discretization shown in these cases, depict the simple Euler integration (either explicit or implicit). Even the Crank-Nicolson scheme is shown like that.

Is my approach correct (discretize spatial derivative and solve using an optimized ode solver)? Can there be numerical stability issues?